EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6139 ISSN 1307-5543 – ejpam.com Published by New York Business Global Trapezoidal and Midpoint-Type Inequalities Based on Extended Conformable Operators Muhammad Samraiz1, Muhammad Qasim1, Gauhar Rahman2,∗, Muhammad Sarwar3,4, Nahid Fatima3, Kamaleldin Abodayeh3 1 Department of Mathematics, University of Sargodha, P.O. Box 40100, Sargodha, Pakistan 2 Department of Mathematics and Statistics, Hazara University, Mansehra 21300, Pakistan 3 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia 4 Department of Mathematics, University of Malakand, Chakdara Dir (Lower), KPK, Pakistan Abstract. In this paper, we explore some inequalities derived from twice differentiable functions together with the extended conformable fractional operators. First, we investigate two lemmas using extended conformable fractional operators. Then, we utilize these results to explore some new trapezoidal and midpoint-type inequalities through the use of the convex property of twice differentiable functions. Moreover, using the power mean inequality and Hölder’s inequality, we introduce a new class of inequalities. The explored results are validated through different 2D and 3D graphs. This new class extends the results of previous research studies. The present paper seeks to motivate researchers to apply these concepts to other fractional operators. 2020 Mathematics Subject Classifications: 26A33, 35J05 Key Words and Phrases: Extended conformable fractional operators, trapezoidal-type inequal- ities, midpoint-type inequalities, Hölder’s inequality, power mean inequality 1. Introduction Convex functions form an essential branch in mathematics with considerable applica- tions across diverse fields. By definition, a convex function lies above the straight line connecting any two points in its domain. The study of convex functions has evolved over the past century, with roots deeply embedded in geometry [1]. Their utility spans vari- ous disciplines, notably physics [2], chemistry [3], medicine [4], optimization, economics, statistics [5], and bioengineering [6]. Additionally, fields such as DC programming [7], ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6139 Email addresses: muhammad.samraiz@uos.edu.pk (M. Samraiz), mqasimsultan191@gmail.com (M. Qasim), gauhar55uom@gmail.com (G. Rahman), sarwarswati@gmail.com (M. Sarwar), nfatima@psu.edu.sa (N. Fatima), kamal@psu.edu.sa (K. Abodayeh) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 2 of 34 convex programming [8], functional analysis [9], monotone operator theory [10], Object Detection Algorithm [11], and complex analysis [12] further underscore the importance of convexity. This characteristic plays a crucial role in solving many real-world problems related to minimizing or maximizing functions subject to certain-constraints. The exploration of convex functions can be traced back to the ancient Greek math- ematician Archimedes (287 BC-212 BC), who, in his work On the Sphere and Cylinder [13], described a convex arc as a curved line in a plane that remains entirely on one side of the straight line connecting its endpoints [14]. The study of mathematical inequalities started in the 18th century with work by Carl Friedrich Gauss. Later, mathematicians like Augustin-Louis Cauchy and Pafnuty Chebyshev explored how inequalities could be used in analysis. A key result came from Viktor Bunyakovsky, who proved an early form of the Cauchy-Schwarz inequality [15]. In the 19th century, Otto Hölder introduced a version of what would later be known as Jensen’s inequality, assuming the second derivative of a function is non-negative. The study of inequalities grew more important in the 20th cen- tury, with major contributions by mathematicians like Leonhard Euler and Adrien-Marie Legendre. In recent years, there has been a growing interest in exploring new aspects of convex functions, particularly in deriving novel inequalities such as Jensen’s inequality [16], the power mean inequality [17], the Cauchy-Schwarz inequality [18], Bell’s inequality [19], Boole’s inequality [20], the Sobolev inequality [21], Chernoff’s inequality [22], the Hermite-Hadamard (H−H) inequality [23], Ostrowski type Inequalities [24], midpoint and trapezoidal-type inequalities [25, 26]. While various types of inequalities exist, convex inequalities play a vital role in this field. As Fractional calculus is a branch of mathemat- ical analysis that generalizes the concepts of differentiation and integration to non-integer order. There are many important applications of fractional calculus, such as modeling influenza [27] and trajectory tracking of the Stanford robot [28]. Mathematicians have derived different operators in fractional calculus to obtain desired results, such as the Riemann-Liouville fractional operators [29], the Caputo-Fabrizio fractional operator [30], the Hilfer fractional derivative operator [31], Hadamard-type fractional operators [32], and Katugampola fractional integrals [33], according to their needs. One such operator is the conformable fractional operator [34], and (k, ρ)−Conformable Fractional Integrals [35]. Moreover mathematicians have made significant efforts to analyze the behavior of inequalities particularly fractional inequalities through computational methods [36]. This work is based on investigating new inequalities of trapezoidal and midpoint type for convex functions. These inequalities are obtained with the aid of the extended con- formable fractional operators and twice differentiable functions. The extended conformable fractional operators generalize the concept of fractional calculus to a wide range of func- tions, and as such, it provides a strong tool for studying inequalities with generalized functions. Also, twice differentiable functions are fundamental in the construction and proof of these inequalities. Additionally, the absolute value function and convex proper- ties of twice differentiable functions are utilized to provide connections between the new inequalities. Also, celebrated inequalities like Hölder inequality [37] and the Power Mean Inequality [37] are utilized in deriving special findings. This paper is organized into sections as follows: Section 2 offers a detailed review of M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 3 of 34 convex functions, including definitions and some properties. Section 3 introduces the new inequalities of trapezoidal type derived by using extended conformable fractional operators along with generalized twice differentiable functions. Section 4 discusses the midpoint type inequalities similar to previous. Finally, Section 5 wraps up the paper by summarizing the findings and proposing avenues for future research. 2. Preliminaries In this section, we discuss fundamental results that will later prove beneficial. The first integrals to be described are the Riemann-Liouville integrals [38]. Additionally, we recall the extended conformable operators [39], which are well documented in the literature and serves as a crucial building block for this paper. Since the beta and gamma functions [40] are key components of fractional calculus, both are discussed here. Definition 1. For positive real numbers ζ and η, the gamma function Γ(ζ) and incomplete beta function β(ζ, η, r) are defined as Γ(ζ) = ∫ ∞ 0 φζ−1e−φ dφ and β(ζ, η, r) = ∫ r 0 φζ−1(1− φ)η−1 dφ, respectively. The Riemann-Liouville fractional integrals [38] are attributed to mathematicians Bern- hard Riemann and Joseph Liouville while Liouville first explored the concept of fractional calculus, Riemann’s work significantly developed the integral operator that is widely used today as the standard definition of a fractional integral. Definition 2. For h ∈ L1[ν, ω], the Riemann-Liouville integrals jαν+h(ζ) and jαω−h(ζ) of order α > 0 are given as jαν+h(ζ) = 1 Γα ∫ ζ ν (ζ − φ)α−1h(φ)dφ, ζ > ν, (1) jαω−h(ζ) = 1 Γα ∫ ω ζ (φ− ζ)α−1h(φ)dφ, ζ < ω, (2) It is evident that setting α = 1 causes the Riemann-Liouville integrals reduce to the stan- dard integrals. Fractional conformable operators [41] were defined by Khalil et al. in 2014. Later ex- tended conformable fractional operators [39] were defined which extend some new concepts to fractional calculus. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 4 of 34 Definition 3. For h ∈ L1[ν, ω], the extended fractional conformable integrals α k j µ ν+h(ζ) and α k j µ ω−h(ζ) of order α ∈ C, Re(α) > 0, k > 0 and µ ∈ (0, 1] are given by α k j µ ν+h(ζ) = 1 kΓk(α) ∫ ζ ν ( (ζ − ν)µ − (φ− ν)µ µ )α k −1 h(φ) (φ− ν)1−µ dφ, ζ > ν (3) and α k j µ ω−h(ζ) = 1 kΓk(α) ∫ ω ζ ( (ω − ζ)µ − (ω − φ)µ µ )α k −1 h(φ) (ω − φ)1−µ dφ, ω > ζ, (4) where Γk(α) = k α k −1Γ(αk ). It can be observed that if k = 1, equations (3) and (4) yield the ordinary conformable operator. Additionally, from usual observations, it can be concluded that when µ = 1 and k = 1, equations (3) and (4) coincide with (1) and (2) respectively. Now, we proceed to define well documented concept of convex function [42] and s- convex function in second sense [43] as fellows. Definition 4. A function h : [a, b] → R is called convex if following inequality holds for all x, y ∈ [a, b] and ρ ∈ [0, 1] h(ρx+ (1− ρ)y) ≤ ρh(x) + (1− ρ)h(y). Definition 5. A function h : [0, ω] → R is said to be s-convex in the second sense if the inequality h(ρx+ (1− ρ)y) ≤ ρsh(x) + (1− ρ)sh(y), holds for all x, y ∈ [0, ω] and ρ, s∈ [0, 1]. This class is usually denoted by K2 s . Let us review Hölder’s and power mean inequalities [37] as follows: Definition 6. Let p > 1 and 1 p + 1 q = 1. If |h|p, |ϕ|q ∈ L[ν, ω] are real functions defined on [ν, ω], then ∫ ω ν |h(x)ϕ(x)|dx ≤ (∫ ω ν |h(x)|pdx ) 1 p (∫ ω ν |ϕ(x)|qdx ) 1 q . Definition 7. For q ≥ 1 and 1 p + 1 q = 1. If h(x), ϕ(x) are real functions defined on [ν, ω] such that |h|p,|ϕ|q ∈ L[ν, ω],then∫ ω ν |h(x)ϕ(x)|dx ≤ (∫ ω ν |h(x)|dx )1− 1 q (∫ ω ν |h(x)||ϕ(x)|qdx ) 1 q . M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 5 of 34 3. Trapezoidal Type Inequalities Based on Extended Conformable Fractional Operators In this section, we derive trapezoidal-type inequalities for twice-differentiable functions. By employing extended conformable fractional operators, we obtain these inequalities. These results extend classical inequalities by incorporating the flexibility and generality of fractional calculus. To establish the trapezoidal-type inequalities for extended conformable fractional operators, we consider the following lemma, which forms the foundation for our subsequent analysis and results. Lemma 1. Consider a function h : [ν, ω] → R that is twice differentiable on (ν, ω) and satisfies h′′ ∈ L1([ν, ω]). In this context, the following equality is established h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α k Γ( α k + 1) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) = (ω − ν)2(kµ) α k η3 [ ∫ 1 0 [∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ] × h′′ ( η − t η ν + t η ω ) dt + ∫ 1 0 [∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ] h′′ ( t η ν + η − t η ω ) dt ] , (5) where ϕk(µ, α) = ω − ν η2 (kµ) α k [ h′ ( η − 1 η ν + 1 η ω ) − h′ ( ν η + η − 1 η ω )] × [∫ 1 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ] . Proof. By considering the integral I1 = ∫ 1 0 [∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ] h′′ ( η − t η ν + t η ω ) dt. Using the technique of integration by parts, it yields I1 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ − η ω − ν ∫ 1 0 ( 1 (kµ) α k − ( 1− (1− t)µ kµ )α k ) h′ ( η − t η ν + t η ω ) dt. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 6 of 34 Again, we apply the technique of integration by parts to the second term; therefore, we obtain the following expression I1 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ + ( η ω − ν )2 h(ν) 1 (ku) α k − ( η ω − ν )2 1 k (α k )∫ 1 0 [ 1− (1− t)µ kµ ]α k −1 (1− t)µ−1 × h ( η − t η ν + t η ω ) dt, (6) Substituting x = η−t η ν + t ηω, we can write I1 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ + ( η ω − ν )2 h(ν) 1 (ku) α k − ( η ω − ν )µα k +2 ( Γ ( α k + 1 ) k.k α k −1Γ(αk ) ) × ∫ (η−1)ν+ω η ν  ( ω−ν η )µ − ( (η−1)ν+ω η − x )µ µ  α k −1 [ (η − 1)ν + ω η − x ]µ−1 dx. By using the relation Γk(α) = k α k −1Γ(αk ), we can write I1 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ + ( η ω − ν )2 h(ν) 1 (ku) α k − ( η ω − ν )µα k +2 Γ( α k + 1). ( 1 kΓkα ) × ∫ (η−1)ν+ω η ν  ( (η−1)ν+ω η − ν )µ − ( (η−1)ν+ω η − x )µ µ  α k −1  h(x)dx[ (η−1)ν+ω η − x ]1−µ  . By using the definition of extended conformable operator (4), we obtain I1 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ + ( η ω − ν )2 h(ν) 1 (ku) α k − ( η ω − ν )µα k +2 Γ (α k + 1 ) α k j µ (η−1)ν+ω) η − h(ν). (7) Similarly I2 = ∫ 1 0 [∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ] h′′ ( t η ν + η − t η ω ) dt, M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 7 of 34 I2 = − η ω − ν h′ ( 1 η ν + η − 1 η ω )∫ 1 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ + ( η ω − ν )2 h(ω) 1 (ku) α k − ( η ω − ν )µα k +2 Γ (α k + 1 )α k jµν+(η−1)ω η + h(ω). (8) Adding equations (7) and (8), we get the required result. Remark 1. If we substitute k=1 and η = 2 in Lemma 5, then obtained result leads to [44, Lemma 3]. Theorem 1. Consider h : [ν, ω] → R as a twice differentiable mapping on (ν, ω) such that h′′ ∈ L([ν, ω]). If |h′′| is s-convex in second sense on [ν, ω] then,∣∣∣∣h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α K Γ( α k + 1) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2(kµ) α k η3 ( |h′′(ν)|+ |h′′(ω)| ηs ) × ∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣ (ts + (η − t)s) dt. (9) Proof. Taking absolute value of equation (5), we have∣∣∣∣h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α k Γ( α k + 1) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2(kµ) α k η3 × [∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣dt + ∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ ) α k ) dφ ∣∣∣∣∣∣∣∣h′′( t η ν + η − t η ω ) ∣∣∣∣dt]. (10) By using s-convexity of |h′′| in second sense, we get∣∣∣∣h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α k Γ( α k + 1) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2(kµ) α k η3 × [∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣ (( t η )s |h′′(ω)|+ ( η − t η )s |h′′(ν)| ) dt M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 8 of 34 + ∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ ) α k ) dφ ∣∣∣∣ (( t η )s |h′′(ν)|+ ( η − t η )s |h′′(ω)| ) dt ] ∣∣∣∣h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α k Γ( α k + 1) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2(kµ) α k η3 × ( |h′′(ν)|+ |h′′(ω)| ηs )∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣ × (ts + (η − t)s) dt. Which lead us to our required result. Remark 2. If we set η = 2,k=1 and s=1 in (9), then inequality reduced to∣∣∣∣h(ν) + h(ω) 2 − 2µα − 1 (ω − ν)µα (µ)αΓ(α+ 1) ( αJµ ν+ω 2 − h(ν) +α Jµ ν+ω 2 + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2µα 8 ( |h′′(ν)|+ |h′′(ω)| ) ∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 µα − ( 1− (1− φ)µ µ )α) dφ ∣∣∣∣dt. (11) Remark 3. When we substitute µ=1 in (11), then Theorem 1 leads to [45], corollary 7. Remark 4. By setting η = 2, k = 1, s = 1, µ = 1 and α = 1 in Theorem 1 then inequality reduced to [46], Proposition 2. Example 1. This example illustrates the verification of the inequality stated in Theorem 1 using graphs and a table. To achieve this, consider the function h(x) = x6+2x4 defined over the interval h : [2, 7] → R. The parameters are chosen as k = 3, α = 4,s=1, and η = 8. Explanation: A 2D plot is created to observe the function’s behavior across the interval µ ∈ (0, 1] shown in Fig. 1, providing a visual way to analyze the inequality (9). To check the validity of the theorem, we proceed to calculate numerical values of the inequality for different µ and k values, which confirm our results. These results are compiled into a table shown in Tables 1 and 2 highlighting the precision and reliability of the inequality. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 9 of 34 Figure 1: This figure illustrates the graphical representation of (9), corresponding to µ ∈ (0, 1] validating our results. Table 1: In this figure summary of values confirming (9) corresponding to µ ∈ (0, 1]. µ 0.2 0.4 0.6 0.8 1 LHS 1368.99 1289.62 1213.3 1142.93 1078.93 RHS 1715.36 1611.45 1512.02 1420.78 1338.12 Table 2: Summary of values confirming (9) for k ∈ [1, 5], while keeping µ = .5 fixed. k 1 2 3 4 5 LHS 1457.94 1365.57 1258.87 1157.15 1065.55 RHS 1775.59 1681.23 1560.81 1441.05 1331.05 To extend the analysis, a 3D plot is generated to evaluate the inequality for parameter ranges α ∈ [5, 10] and µ ∈ (0, 1]. Figure 2 displays this 3D visualization, offering further support for the theorem. Figure 2: In Fig. 2 three dimensional visualization validating the inequality of Theorem 1 corresponding to µ ∈ (0, 1] and α ∈ [5, 10]. The combination of these methods confirms the robustness and applicability of Theorem 1 in describing the behavior of h(x) under the given conditions. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 10 of 34 Theorem 2. Assume that h : [ν, ω] → R is a twice differentiable function on (ν, ω) such that h′′ ∈ Lp([ν, ω]) with ν < ω. Let |h′′|q be s-convex in second sense on [ν, ω] with q > 1 then inequality is given by,∣∣∣∣h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α k Γ (α k + 1 ) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2(kµ) α k η3 × (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣pdt ) 1 p × ([( |h′′(ω)|q ηs(s+ 1) − |h′′(ν)|q ηs(s+ 1) )( (η − 1)s+1 − ηs+1 )] 1 q + [( |h′′(ν)|q ηs(s+ 1) − |h′′(ω)|q ηs(s+ 1) )( (η − 1)s+1 − ηs+1 )] 1 q ) . (12) Proof. By employing Hölder inequality on (10), we get∣∣∣∣h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α K Γ( α k + 1) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2(kµ) α k η3 × [(∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣pdt ) 1 p × (∫ 1 0 ∣∣∣∣h′′((η − t η ) ν + t η ω ) ∣∣∣∣qdt) 1 q + (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣pdt ) 1 p × (∫ 1 0 ∣∣∣∣h′′( t η ν + ( η − t η ) ω ) ∣∣∣∣qdt) 1 q . (13) By considering s-convexity in second sense of |h′′(x)|q, then∫ 1 0 ∣∣∣∣h′′((η − t η ) ν + t η ω ) ∣∣∣∣qdt ≤ ∫ 1 0 [( η − t η )s |h′′(ν)|q + |h′′(ω)|q ( t η )s] dt ≤ |h′′(ω)|q ηs(s+ 1) − |h′′(ν)|q ηs(s+ 1) [ (η − 1)s+1 − ηs+1 ] . (14) Similarly∫ 1 0 ∣∣∣∣h′′( t η ν + ( 1− t η ) ω ) ∣∣∣∣qdt ≤ |h′′(ν)|q ηs(s+ 1) − |h′′(ω)|q ηs(s+ 1) [ (η − 1)s+1 − ηs+1 ] . (15) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 11 of 34 Substituting both (14) and (15) in (13), we conclude∣∣∣∣h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α k Γ (α k + 1 ) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2(kµ) α k η3 × (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣pdt ) 1 p × ([( |h′′(ω)|q ηs(s+ 1) − |h′′(ν)|q ηs(s+ 1) )( (η − 1)s+1 − ηs+1 )] 1 q + [( |h′′(ν)|q ηs(s+ 1) − |h′′(ω)|q ηs(s+ 1) )( (η − 1)s+1 − ηs+1 )] 1 q ) . So inequality is proved. Remark 5. If we substitute η = 2, k = 1 and s=1 in equation (12), then following result is obtained∣∣∣∣h(ν) + h(ω) 2 − 2µα−1 (ω − ν)µα µαΓ(α+ 1) ( αJµ ν+ω 2 − h(ν) +α Jµ ν+ω 2 + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2µα 8 (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (µ)α − ( 1− (1− φ)µ µ )α) dφ ∣∣∣∣pdt) 1 p [( 3|h′′(ν)|q + |h′′(ω)|q 4 ) 1 q + ( |h′′(ν)|q + 3|h′′(ω)|q 4 ) 1 q ] . (16) Corollary 1. If we consider µ = α = 1 in (16) we arrive at the conclusion that is∣∣∣∣h(ν) + h(ω) 2 − 1 (ω − ν) ∫ ω ν h(x)dx ∣∣∣∣ ≤ (ω − ν)2 8 ( 1 p+ 1 − 1 2p(2p+ 1) ) 1 p [( 3|h′′(ν)|q + |h′′(ω)|q 4 ) 1 q + ( |h′′(ν)|q + 3|h′′(ω)|q 4 ) 1 q ] . Proof. Substituting value µ = α = 1 in (16),∣∣∣∣h(ν) + h(ω) 2 − 1 (ω − ν) Γ(α+ 1) ∫ ω ν h(x)dx ∣∣∣∣ ≤ (ω − ν)2µα 8 (∫ 1 0 ∣∣∣∣ ∫ t 0 (1− φ) dφ ∣∣∣∣pdt) 1 p × [( 3|h′′(ν)|q + |h′′(ω)|q 4 ) 1 q + ( |h′′(ν)|q + 3|h′′(ω)|q 4 ) 1 q ] . (17) Consider (∫ 1 0 ∣∣∣∣ ∫ t 0 (1− φ)dφ ∣∣∣∣pdt) 1 p = (∫ 1 0 ∣∣∣∣t− t2 2 ∣∣∣∣pdt) 1 p . M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 12 of 34 Under condition |A−B|p ≤ Ap −Bp when A > B > 0; and p > 1 so we can write(∫ 1 0 ∣∣∣∣ ∫ t 0 (1− φ)dφ ∣∣∣∣pdt) 1 p ≤ (∫ 1 0 tpdt− ∫ 1 0 t2p 2p dt ) 1 p (∫ 1 0 ∣∣∣∣ ∫ t 0 (1− φ)dφ ∣∣∣∣pdt) 1 p ≤ [ 1 p+ 1 − 1 2p(2p+ 1) ] 1 p . (18) Substituting (18) in (17), we get our desired result. Example 2. This example illustrates the applicability of Theorem 2 through graphical and numerical analyses. We consider the function h(x) = x6 + 2x4, defined on the interval [2, 7], and verify the inequality using specific parameter values: k = 3, α = 4, s = 1, 1 p = .6, 1q = .4, and η = 8. Explanation: Figure 3 presents a 2D graph of the inequality over µ ∈ (0, 1], showing the behavior of both the left-hand side and the right-hand side of inequality (12). The plot demonstrates that the L.H.S remains within the bounds prescribed by the R.H.S, visually validating the theorem. A numerical analysis is performed by computing the L.H.S. and R.H.S. Figure 3: 2D plot for µ ∈ (0, 1] of Theorem 2 supporting the validation of theorem. values for several instances of µ with k fixed, as shown in Table 3. Additionally, by fixing µ = 0.5, a table is constructed for different values of k, as shown in Table 4. This tabular representation is highlighting the accuracy and consistency of the inequality. Table 3: Summary of inequality (12) confirms its validity for µ ∈ (0, 1]. µ 0.2 0.4 0.6 0.8 1 LHS 1368.99 1289.62 1213.3 1142.93 1078.93 RHS 2448.47 2282.73 2203.15 1986.66 1861.13 M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 13 of 34 Table 4: Summary of inequality (12) confirms its validity for k ∈ [1, 5] while fixing µ = .5. µ 1 2 3 4 5 LHS 1415.4 1343.89 1250.8 1157.15 1070.49 RHS 2543.26 2389.52 2203.15 2023.35 1861.33 To extend the analysis, we explore the inequality in three dimensions by varying the parameters α ∈ [5, 10] and µ ∈ (0, 1]. Figure 4 illustrates the resulting surface, further confirming the robustness of the inequality across a range of parameter values. These Figure 4: 3D representation verifying the inequality in Theorem 2. combined analyses establish the reliability and practical utility of Theorem 2 in bounding the behavior of h(x) under the given conditions. Theorem 3. Consider h:[ν, ω] → R as a twice differentiable mapping on (ν, ω) such that h′′ ∈ Lq([ν, ω]). Assume that |h′′| admits the s-convexity in second sense on [ν, ω] with q ≥ 1 then, ∣∣∣∣h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α k Γ( α k + 1) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2(kµ) α k η3 × (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣dt )1− 1 q (19) × ([ |h′′(v)|qΨ2(µ, α) + |h′′(w)|qΨ1(µ, α) ns ] 1 q + [ |h′′(v)|qΨ1(µ, α) + |h′′(w)|qΨ2(µ, α) ns ] 1 q ) , where M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 14 of 34 Ψ1(µ, α) = ∫ 1 0 ts ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣dt, Ψ2(µ, α) = ∫ 1 0 (η − t)s ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣dt. Proof. By employing power mean inequality on equation (10), we obtain∣∣∣∣h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α k Γ( α k + 1) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2(kµ) α k η3 × [(∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣dt )1− 1 q × (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt ) 1 q + (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣dt )1− 1 q × (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ ) α k ) dφ ∣∣∣∣∣∣∣∣h′′( t η ν + η − t η ω ) ∣∣∣∣qdt) 1 q ] . (20) By taking advantage of s-convexity of |h′′| on the following terms, we proceed as(∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt ) ≤ ∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣ ((η − t η )s |h′′(ν)|q + ( t η )s |h′′(ω)|q ) dt (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt ) ≤ |h′′(ν)|q ηs (∫ 1 0 (η − t)s ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣dt) + |h′′(ω)|q ηs (∫ 1 0 ts ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣dt ) . (21) Assuming Ψ1(µ, α) = ∫ 1 0 ts ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣dt, (22) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 15 of 34 Ψ2(µ, α) = ∫ 1 0 (η − t)s ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣dt. (23) Substituting equations (22) and (23) in (21)(∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt ) ≤ ( |h′′(ν)|q ηs Ψ2(µ, α) + |h′′(ω)|q ηs Ψ1(µ, α) ) . (24) Similarly (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt ) ≤ ( |h′′(ν)|q ηs Ψ1(µ, α) + |h′′(ω)|q ηs Ψ2(µ, α) ) . (25) Substituting (24) and (25) in (20)∣∣∣∣h(ν) + h(ω) η + ϕk(µ, α)− η µα k −1 (ω − ν) µα k (kµ) α k Γ( α k + 1) × ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2(kµ) α k η3 × (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1 (kµ) α k − ( 1− (1− φ)µ kµ )α k ) dφ ∣∣∣∣dt )1− 1 q × ([ |h′′(v)|qΨ2(µ, α) + |h′′(w)|qΨ1(µ, α) ns ] 1 q + [ |h′′(v)|qΨ1(µ, α) + |h′′(w)|qΨ2(µ, α) ns ] 1 q ) . (26) Consequently, we obtain the desired outcome. Corollary 2. If we substitute k = 1, s = 1 and µ = 1 in (26), then∣∣∣∣h(ν) + h(ω) η + ϕ1(1, α)− ηα−1 (ω − ν)µα Γ(α+ 1) ( αJ (η−1)ν+ω η − h(ν) +α J ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2 η3 ( 1 2 − 1 (α+ 1)(α+ 3) )1− 1 q × [(( η ( 1 2 − 1 (α+ 1)(α+ 2) ) − ( 1 3 − 1 (α+ 1)(α+ 3) )) |h′′(ν)|q η + ( 1 3 − 1 (α+ 1)(α+ 3) ) |h′′(ω)|q η ) 1 q + (( 1 3 − 1 (α+ 1)(α+ 3) ) |h′′(ν)|q η M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 16 of 34 + ( η ( 1 2 − 1 (α+ 1)(α+ 2) ) − ( 1 3 − 1 (α+ 1)(α+ 3) )) |h′′(ω)|q η ) 1 q ] . (27) Proof. Setting k = µ = 1 in (22), Ψ1(1, α) = ∫ 1 0 ∣∣∣∣ ∫ t 0 (1− φα) dφ ∣∣∣∣dt Ψ1(1, α) = ∫ 1 0 |t− tα+1 α+ 1 |dt When A > B then |A−B| = A−B Ψ1(1, α) = 1 2 − 1 (α+ 1)(α+ 2) (28) Similarly Ψ2(1, α) = 1 3 − 1 (α+ 1)(α+ 3) (29) Substituting (28) and (29) in (26),∣∣∣∣h(ν) + h(ω) η + ϕ1(1, α)− ηα−1 (ω − ν)α Γ(α+ 1) ( αJ (η−1)ν+ω η − h(ν) +α J ν+(η−1)ω η + h(ω) ) ∣∣∣∣ ≤ (ω − ν)2 η3 ( 1 2 − 1 (α+ 1)(α+ 3) )1− 1 q × [(( η ( 1 2 − 1 (α+ 1)(α+ 2) ) − ( 1 3 − 1 (α+ 1)(α+ 3) )) |h′′(ν)|q η + ( 1 3 − 1 (α+ 1)(α+ 3) ) |h′′(ω)|q η ) 1 q + (( 1 3 − 1 (α+ 1)(α+ 3) ) |h′′(ν)|q η + ( η ( 1 2 − 1 (α+ 1)(α+ 2) ) − ( 1 3 − 1 (α+ 1)(α+ 3) )) |h′′(ω)|q η ) 1 q ] . (30) Ultimately, the expected outcome has been reached. Remark 6. If we assign η = 2 and α = 1 in (27), our result reduced as follow∣∣∣∣h(ν) + h(ω) 2 − 1 ω − ν ∫ ω ν h(x)dx ∣∣∣∣ ≤ (ω − ν)2 24 [( 11 16 |h′′(ν)|q + 5 16 |h′′(ω)|q ) 1 q + ( 5 16 |h′′(ν)|q + 11 16 |h′′(ω)|q ) 1 q ] . Example 3. This example demonstrates the application of Theorem 3 through both graph- ical and numerical methods. We analyze the function h(x) = x6 + 2x4 within the interval [2, 7] to verify the inequality using the following parameter values: k = 3, α = 4, 1 p = 0.6, M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 17 of 34 1 q = 0.4, and η = 8. Explanation: Figure 5 displays a two-dimensional plot illustrating the inequality for µ ∈ (0, 1]. The graph of inequality (19) is clearly showing that the LHS consistently stays within the bounds set by the RHS. Numerical calculations are performed to compare the inequality at various values of µ. These results are summarized in the table provided in Fig. 5, highlighting the precision and consistency of the inequality. Figure 5: This figure provides a visualization of Theorem 3 using a 2D graph confirming its validity. Table 5: Table shown the summary of equation (19) corresponding to µ ∈ (0, 1] as evidence of inequality. µ 0.2 0.4 0.6 0.8 1 LHS 1368.99 1289.62 1213.3 1142.93 1078.93 RHS 2271.79 2132.02 1998.51 1876.17 1765.5 To further validate the results, the inequality is analyzed in three dimensions by varying α ∈ [5, 10] and µ ∈ (0, 1]. Figure 6 shows a 3D surface plot, confirming the robustness of the inequality across a broader parameter range. Figure 6: 3D visualization validating the inequality in Theorem 3. Together, these analyses reinforce the accuracy and practical relevance of Theorem 3 in characterizing the behavior of h(x) under the specified conditions. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 18 of 34 4. Mid Point Type Inequalities Based on Extended Conformable Fractional Operators In this section, we delve into the study of midpoint inequalities derived from twice- differentiable functions. By employing extended conformable fractional operators and twice-differentiable operators, we aim to establish new insights and results in this area. Our approach begins with the formulation of a fundamental identity, which serves as the foundation for deriving the desired inequalities.By leveraging the capabilities of extended conformable fractional operators, we extend the applicability of midpoint inequalities to a broader class of functions, offering a more comprehensive understanding of their behavior and potential applications. The results obtained in this study not only contribute to the theoretical framework but also pave the way for future research in related domains. Lemma 2. Let h : [ν, ω] → R be a twice differentiable mapping on (ν, ω) with h′′ ∈ L1([ν, ω]). Then the following equality holds: η µα k −1 (ω − ν) µα k (kµ) α k Γ( α k + 1) ( α kJ µ (η−1)ν+ω η − h(ν) +α k Jµ ν+(η−1)ω η + h(ω) ) +Υk[µ, α] − 1 η [ h ( η − 1 η ν + 1 η ω ) + h ( 1 η ν + η − 1 η ω )] = (ω − ν)2(kµ) α k η3 [∫ 1 0 [∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ] h′′ ( η − t η ν + t η ω ) dt + ∫ 1 0 [∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ] h′′ ( t η ν + η − t η ω ) dt ] , (31) where Υk[µ, α] = (ω − ν) η2 (kµ) α k ( h′ ( η − 1 η ν + 1 η ω ) − h′ ( 1 η ν + η − 1 η ω )) × [∫ 1 0 ( 1− (1− φ)µ kµ )α k dφ ] . Proof. Consider I3 = ∫ 1 0 [∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ] h′′ ( η − t η ν + t η ω ) dt, Utilizing the technique of integration by parts, we obtain I3 = η ω − ν h′ ( η − t η ν + t η ω )∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣1 0 − η ω − ν ∫ 1 0 ( 1− (1− t)µ kµ )α k h′ ( η − t η ν + t η ω ) dt, M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 19 of 34 I3 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1− (1− φ)µ kµ )α k dφ − η ω − ν ∫ 1 0 ( 1− (1− t)µ kµ )α k h′ ( η − t η ν + t η ω ) dt, Again employing integration by parts on second term I3 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1− (1− φ)µ kµ )α k dφ − ( η ω − ν )2(1− (1− t)µ kµ )α k h ( η − t η ν + t η ω ) ∣∣∣∣1 0 + ( η ω − ν )2 . 1 k (α k )∫ 1 0 [ 1− (1− t)µ kµ ]α k −1 (1− t)µ−1 × h ( η − t η ν + t η ω ) dt I3 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1− (1− φ)µ kµ )α k dφ− ( η ω − ν )2( 1 kµ )α k h ( η − 1 η ν + 1 η ω ) + ( η ω − ν )2 . 1 k 1 k α k −1 (α k )∫ 1 0 [ 1− (1− t)µ kµ ]α k −1 (1− t)µ−1 × h ( η − t η ν + t η ω ) dt. Substituting x = η−t η ν + t ηω, we can write I3 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1− (1− φ)µ kµ )α k dφ − ( m ω − ν )2( 1 kµ )α k h ( η − 1 η ν + 1 η ω ) + ( η ω − ν )µα k +2 ( Γ ( α k + 1 ) k.k α k −1Γ(αk ) ) × ∫ (η−1)ν+ω η ν  ( ω−ν η )µ − ( (η−1)ν+ω η − x )µ µ  α k −1 [ (η − 1)ν + ω η − x ]µ−1 h(x)dx. By using relation Γkα = k α k −1Γ ( α k ) , we can write I3 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1− (1− φ)µ kµ )α k dφ− ( η ω − ν )2( 1 kµ )α k h ( η − 1 η ν + 1 η ω ) + ( η ω − ν )µα k +2 Γ( α k + 1) ( 1 kΓkα )∫ (η−1)ν+ω η ν  ( (η−1)ν+ω η − ν )µ − ( (η−1)ν+ω η − x )µ µ  α k −1 (32) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 20 of 34 ×  h(ν)dx[ (η−1)ν+ω η − x ]1−µ  . (33) By using the definition of extended conformable operator (4), we can write I3 = η ω − ν h′ ( η − 1 η ν + 1 η ω )∫ 1 0 ( 1− (1− φ)µ kµ )α k dφ − ( η ω − ν )2( 1 kµ )α k h ( η − 1 η ν + 1 η ω ) + ( η ω − ν )µα k +2 Γ( α k + 1)αk j µ (η−1)ν+ω η − h(ν). (34) Similarly I4 = ∫ 1 0 [∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ] h′′ ( t η ν + η − t η ω ) dt I4 = − η ω − ν h′ ( 1 η ν + η − 1 η ω )∫ 1 0 ( 1− (1− φ)µ kµ )α k dφ − ( η ω − ν )2 h ( ν η + η − 1 η ω ) 1 (ku) α k + ( η ω − ν )µα k +2 Γ (α k + 1 )α k jµν+(η−1)ω η + h(ω). (35) Adding equations (34) and (35), we get our required result. Remark 7. By considering η = 2 and k=1 simultaneously in equation (31), that directly leads to [[44], Lemma 10]. Theorem 4. Assume that h : [ν, ω] → R as a twice differentiable function on (ν, ω) such that h′′ ∈ L1([ν, ω]). By considering the convexity |h′′| on [ν, ω] the inequality is given as,∣∣∣∣ ηµ α k −1 (ω − ν)µ α k (kµ) α k Γ (α k + 1 )( α k j µ (η−1)ν+ω η − h(ν) +α k jµν+(η−1)ω η + h(ω) ) +Υk[µ, α] − 1 η ( h ( η − 1 η ν + 1 η ω ) + h ( 1 η ν + η − 1 η ω )) ∣∣∣∣ ≤ (ω − ν)2 η3 (kµ) α k 1 µ ∫ 1 0 ∣∣∣∣ 1 (kµ) α k B ( α k + 1, 1 µ , 1− (1− t)µ ) ∣∣∣∣(ts + (η − t)s) dt × ( |h′′(ν)|+ |h′′(ω)| ηs ) , (36) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 21 of 34 where Υk[µ, α] = (ω − ν) η2 (kµ) α k ( h′ ( η − 1 η ν + 1 η ω ) − h′ ( 1 η ν + η − 1 η ω )) × [∫ 1 0 ( 1− (1− φ)µ kµ )α k dφ ] . Proof. Taking absolute value on both sides of equation (31)∣∣∣∣ ηµ α k −1 (ω − ν)µ α k (kµ) α k Γ (α k + 1 )( α k j µ (η−1)ν+ω η − h(ν) +α k jµν+(η−1)ω η + h(ω) ) +Υk[µ, α] − 1 η ( h ( η − 1 η ν + 1 η ω ) + h ( 1 η ν + η − 1 η ω )) ∣∣∣∣ ≤ (ω − ν)2 η3 (kµ) α k [∫ 1 0 ∣∣∣∣∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣dt + ∫ 1 0 ∣∣∣∣∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣∣∣∣∣h′′( t η ν + η − t η ω ) ∣∣∣∣dt]. (37) By considering s-convexity of |h′′| in second sense∣∣∣∣ ηµ α k −1 (ω − ν)µ α k (kµ) α k Γ (α k + 1 )( α k j µ (η−1)ν+ω η − h(ν) +α k jµν+(η−1)ω η + h(ω) ) +Υk[µ, α] − 1 η ( h ( η − 1 η ν + 1 η ω ) + h ( 1 η ν + η − 1 η ω )) ∣∣∣∣ ≤ (ω − ν)2 η3 (kµ) α k [∫ 1 0 ∣∣∣∣∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣((η − t η )s |h′′(ν)|+ ( t η )s |h′′(ω)| ) dt + ∫ 1 0 ∣∣∣∣∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣(( t η )s |h′′(ν)|+ ( η − t η )s |h′′(ω)| ) dt ] ∣∣∣∣ ηµ α k −1 (ω − ν)µ α k (kµ) α k Γ (α k + 1 )( α k j µ (η−1)ν+ω η − h(ν) +α k jµν+(η−1)ω η + h(ω) ) +Υk[µ, α] − 1 η ( h ( η − 1 η ν + 1 η ω ) + h ( 1 η ν + η − 1 η ω )) ∣∣∣∣ (38) ≤ (ω − ν)2 η3 (kµ) α k ∫ 1 0 ∣∣∣∣∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣(ts + (η − t)s) dt × ( |h′′(ν)|+ |h′′(ω)| ηs ) . (39) Consider ϖk[µ, α] = ∣∣∣∣∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣ M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 22 of 34 Assuming p = 1− (1− φ)µ dφ = 1 µ (1− p) 1 µ −1 dp ϖk[µ, α] = 1 µ ∣∣∣∣ 1 (kµ) α k ∫ 1−(1−t)µ 0 p( α k +1)−1(1− p) 1 µ −1 dp ∣∣∣∣ By using definition of incomplete beta function , we can write ϖk[µ, α] = 1 µ ∣∣∣∣ 1 (kµ) α k B ( α k + 1, 1 µ , 1− (1− t)µ ) ∣∣∣∣. (40) Substituting (40) in equation (38), we obtained ≤ (ω − ν)2 η3 (kµ) α k ∫ 1 0 1 µ ∣∣∣∣ 1 (kµ) α k B ( α k + 1, 1 µ , 1− (1− t)µ ) ∣∣∣∣(ts + (η − t)s) dt × ( |h′′(ν)|+ |h′′(ω)| ηs ) . (41) Ultimately, the expected outcome has been reached. Remark 8. If we set parameters as η = 2, s = 1 and k=1 in (36), then Theorem 4 leads to [44] Theorem 11. Remark 9. If we set η = 2, s = 1, k = 1 and µ = 1 in (36) then our result reduced to [46] Theorem 1.5. Remark 10. If we set η = 2, s = 1, k = 1, µ = 1 and α = 1 simultaneously in 36, then Theorem 4 and [44], Proposition 1 become identical. Example 4. This example demonstrates the application of Theorem 4 using both graphical and numerical methods. We consider the function h(x) = x6+2x4, defined on the interval [2, 7], and evaluate the inequality under specific parameter values: k = 3,s = 1 α = 4, and η = 8. Explanation: Figure 7 presents a 2D plot illustrating the behavior of both sides of the inequality (36) as µ varies within (0, 1]. The graph confirms that the left-hand side remains within the bounds set by the right-hand side, visually supporting the theorem’s validity. Numerical computations were performed to compare LHS and RHS values for selected values of µ. A table summarizing these results is shown in Figure 6, demonstrating consistency between the computed values and the inequality constraints. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 23 of 34 Figure 7: This figure provides a graphical illustration of Theorem 4 for µ ∈ (0, 1]. Table 6: The table displays numerical values corresponding to equation (36) for µ ∈ (0, 1] which is supporting our results. µ 0.2 0.4 0.6 0.8 1 LHS 65.30 144.68 221 291.37 355.37 RHS 85.95 189.87 289.29 380.54 463.19 Extending the analysis, a 3D visualization was generated by varying α over [5, 10] and µ within (0, 1]. The resulting surface plot, shown in Figure 8, demonstrates that the inequality holds robustly across these parameter ranges. This multi-faceted approach Figure 8: Three-dimensional representation of Theorem 4 validating the inequality across varying α and µ. highlights the validity and practical applicability of Theorem 4, confirming its effectiveness in bounding the behavior of h(x) under the specified conditions. Theorem 5. Assume h : [ν, ω] → R is a twice continuously differentiable function over (ν, ω), with h′′ ∈ L1([ν, ω]). Let |h′′|q be convex on [ν, ω], where q > 1, then∣∣∣∣ ηµ α k −1 (ω − ν)µ α k (kµ) α k Γ (α k + 1 )( α k j µ (η−1)ν+ω η − h(ν) +α k jµν+(η−1)ω η + h(ω) ) +Υk[µ, α] − 1 η ( h ( η − 1 η ν + 1 η ω ) + h ( 1 η ν + η − 1 η ω )) ∣∣∣∣≤ (ω − ν)2 η3 (kµ) α k × [ ∫ 1 0 ∣∣∣∣∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣pdt] 1 p [( |h′′(ω)|q ηs(s+ 1) − |h′′(ν)|q ηs(s+ 1) ( (η − 1)s+1 − ηs+1 )) 1 q M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 24 of 34 + ( |h′′(ν)|q ηs(s+ 1) − |h′′(ω)|q ηs(s+ 1) ( (η − 1)s+1 − ηs+1 )) 1 q ] . (42) Proof. By employing Hölder inequality on equation (37)∣∣∣∣ ηµ α k −1 (ω − ν)µ α k (kµ) α k Γ (α k + 1 )( α k j µ (η−1)ν+ω η − h(ν) +α k jµν+(η−1)ω η + h(ω) ) +Υk[µ, α] − 1 η ( h ( η − 1 η ν + 1 η ω ) + h ( 1 η ν + η − 1 η ω )) ∣∣∣∣≤ (ω − ν)2 η3 (kµ) α k × [(∫ 1 0 ∣∣∣∣∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣pdt ) 1 p ∣∣∣∣(∫ 1 0 ∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt) 1 q + (∫ 1 0 ∣∣∣∣∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣pdt ) 1 p (∫ 1 0 ∣∣∣∣h′′( t η ν + η − t η ω ) ∣∣∣∣qdt) 1 q . (43) By taking advantage of s-convexity of |h′′(x)|q on following equations, consider(∫ 1 0 ∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt) 1 q ≤ (∫ 1 0 [( η − t η )s |h′′(ν)|q + ( t η )s |h′′(ω)|q ] dt ) 1 q (∫ 1 0 ∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt) 1 q ≤ ( |h′′(ω)|q 1 ηs ( ts+1 s+ 1 ) ∣∣∣∣1 0 − |h′′(ν)|q ηs(s+ 1) (η − t)s+1 ∣∣∣∣1 0 ) 1 q (∫ 1 0 ∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt) 1 q ≤ ( |h′′(ω)|q ηs(s+ 1) − |h′′(ν)|q ηs(s+ 1) ( (η − 1)s+1 − ηs+1 )) 1 q . (44) Similarly(∫ 1 0 ∣∣∣∣h′′( t η ν + η − t η ω ) ∣∣∣∣qdt) 1 q ≤ ( |h′′(ν)|q ηs(s+ 1) − |h′′(ω)|q ηs(s+ 1) ( (η − 1)s+1 − ηs+1 )) 1 q . (45) Substituting (44) and (45) in (43), we get∣∣∣∣ ηµ α k −1 (ω − ν)µ α k (kµ) α k Γ (α k + 1 )( α k j µ (η−1)ν+ω η − h(ν) +α k jµν+(η−1)ω η + h(ω) ) +Υk[µ, α] − 1 η ( h ( η − 1 η ν + 1 η ω ) + h ( 1 η ν + η − 1 η ω )) ∣∣∣∣≤ (ω − ν)2 η3 (kµ) α k × [ ∫ 1 0 ∣∣∣∣∫ t 0 ( 1− (1− φ)µ kµ )α k dφ ∣∣∣∣pdt] 1 p [( |h′′(ω)|q ηs(s+ 1) − |h′′(ν)|q ηs(s+ 1) ( (η − 1)s+1 − ηs+1 )) 1 q + ( |h′′(ν)|q ηs(s+ 1) − |h′′(ω)|q ηs(s+ 1) ( (η − 1)s+1 − ηs+1 )) 1 q ] . (46) Therefore, the calculations produce the anticipated result. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 25 of 34 Remark 11. By substituting k=1,s=1and η = 2 in (42) we conclude∣∣∣∣ 2µα−1 (ω − ν)µα (µ)αΓ (α+ 1) ( αjµν+ω 2 −h(ν) + α jµν+ω 2 + h(ω) ) − h ( ν + ω 2 ) ∣∣∣∣ ≤ (ω − ν)2 8 µα [ ∫ 1 0 ∣∣∣∣∫ t 0 ( 1− (1− φ)µ µ )α dφ ∣∣∣∣pdt] 1 p [( 3|h′′(ν)|q + |h′′(ω)|q 4 ) 1 q + ( |h′′(ν)|q + 3|h′′(ω)|q 4 ) 1 q ] . (47) Remark 12. By taking values µ = 1 in (47), result reduced to∣∣∣∣ 2α−1 (ω − ν)α Γ (α+ 1) ( αj ν+ω 2 −h(ν) + α j ν+ω 2 +h(ω) ) − h ( ν + ω 2 ) ∣∣∣∣ ≤ (ω − ν)2 8 [ ∫ 1 0 ∣∣∣∣∫ t 0 sαdφ ∣∣∣∣pdt] 1 p [( 3|h′′(ν)|q + |h′′(ω)|q 4 ) 1 q + ( |h′′(ν)|q + 3|h′′(ω)|q 4 ) 1 q ] . (48) Consider the term [ ∫ 1 0 ∣∣∣∣∫ t 0 sαdφ ∣∣∣∣pdt] 1 p = [ ∫ 1 0 ∣∣∣∣ tα+1 α+ 1 ∣∣∣∣pdt] 1 p [ ∫ 1 0 ∣∣∣∣∫ t 0 sαdφ ∣∣∣∣pdt] 1 p = ( 1 (α+ 1)p ∫ 1 0 tpα+pdt ) 1 p [ ∫ 1 0 ∣∣∣∣∫ t 0 sαdφ ∣∣∣∣pdt] 1 p = ( 1 (α+ 1)p(pα+ p+ 1) ) 1 p ∣∣∣∣ 2α−1 (ω − ν)α Γ(α+ 1) ( αj ν+ω 2 −h(ν) + α j ν+ω 2 +h(ω) ) − h ( ν + ω 2 ) ∣∣∣∣ ≤ (ω − ν)2 8 ( 1 (α+ 1)p(pα+ p+ 1) ) 1 p [( 3|h′′(ν)|q + |h′′(ω)|q 4 ) 1 q + ( |h′′(ν)|q + 3|h′′(ω)|q 4 ) 1 q ] . Remark 13. Assuming α = 1 in (48), we obtain following result∣∣∣∣ 1 ω − ν ∫ ω ν h(x)dx− h ( ν + ω 2 ) ∣∣∣∣≤ (ω − ν)2 16 ( 1 2p+ 1 ) 1 p × [( 3|h′′(ν)|q + |h′′(ω)|q 4 ) 1 q + ( |h′′(ν)|q + 3|h′′(ω)|q 4 ) 1 q ] . (49) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 26 of 34 Example 5. This example illustrates the use of Theorem 5 through a combination of graphical and numerical techniques. The function under consideration is h(x) = x6+2x4, defined on the interval [2, 7], with the inequality evaluated using the parameters k = 3, α = 4,s=1, 1 p = 0.6, 1 q = 0.4, and η = 8. Explanation: Figure 9 provides a 2D visualization of the inequality (42), depicting the behavior of the left-hand side and right-hand side as µ varies in (0, 1]. The graph demonstrates that the LHS consistently adheres to the bounds imposed by the RHS, supporting the validity of the theorem. To complement the graphical analysis, numerical evaluations of the LHS and RHS were conducted for specific values of µ. The results, displayed in a Table 7, confirm that the inequality holds under the given parameters. Figure 9: This figure is illustrating the 2D visualization of the inequality (42) confirming the validity its validity. Table 7: This table summarizes the numerical results of (42). µ 0.2 0.4 0.6 0.8 1 LHS 65.30 144.68 221 291.37 355.37 RHS 151.38 328.386 492.93 640.33 771.12 Further validation was performed by generating a 3D representation of the inequality as α ranges within [5, 10] and µ within (0, 1]. The resulting surface plot, presented in Figure 10, illustrates that the inequality remains valid across the explored parameter space. This Figure 10: Three-dimensional validation of Theorem 5 across different values of α and µ. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 27 of 34 comprehensive analysis underscores the utility and reliability of Theorem 5, verifying its capacity to constrain the behavior of h(x) under the specified conditions. Theorem 6. Let h : [ν, ω] → R be a twice differentiable mapping on (ν, ω) such that h′′L1([ν, ω]). Let |h′′|q be a s-convex in second sense on [ν, ω] with q > 1 then,∣∣∣∣ ηµ α k −1 (ω − ν)µ α k (kµ) α k Γ (α k + 1 )( α k j µ (η−1)ν+ω η − h(ν) +α k jµν+(η−1)ω η + h(ω) ) +Υk[µ, α] − 1 η ( h ( η − 1 η ν + 1 η ω ) + h ( 1 η ν + η − 1 η ω )) ∣∣∣∣≤ (ω − ν)2 η3 (kµ) α k × (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣dt )1− 1 q [( |h′′(ν)|q ηs ξ2k(µ, α) + |h′′(ω)|q ηs ξ1k(µ, α) ) 1 q + ( |h′′(ν)|q ηs ξ1k(µ, α) + |h′′(ω)|q ηs ξ2k(µ, α) ) 1 q ] (50) where ξ1k(µ, α) = ∫ 1 0 ts ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣dt, ξ2k(µ, α) = ∫ 1 0 (η − t)s ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣dt. Proof. By employing Power mean inequality on equation (37),∣∣∣∣ ηµ α k −1 (ω − ν)µ α k (kµ) α k Γ (α k + 1 )( α k j µ (η−1)ν+ω η − h(ν) +α k jµν+(η−1)ω η + h(ω) ) +Υk[µ, α] − 1 η ( h ( η − 1 η ν + 1 η ω ) + h ( 1 η ν + η − 1 η ω )) ∣∣∣∣≤ (ω − ν)2 η3 (kµ) α k × [(∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣dt )1− 1 q × (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣.∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt) 1 q + (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣dt )1− 1 q × (∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣.∣∣∣∣h′′( t η ν + η − t η ω ) ∣∣∣∣qdt ) 1 q ] . (51) By considering s-convexity of |h′′(x)|q in second sense, consider∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣.∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 28 of 34 ≤ ∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣ (|h′′(ν)|(η − t η )s + ( t η )s |h′′(ω)| ) dt ∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣.∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt ≤ |h′′(ν)|q ηs ∫ 1 0 (η − t)s ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣dt + |h′′(ω)|q ηs ∫ 1 0 ts ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣dt. (52) Assuming ξ1k(µ, α) = ∫ 1 0 ts ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣dt, (53) ξ2k(µ, α) = ∫ 1 0 (η − t)s ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣dt. (54) Substituting (53) and (54) in (52), we obtain∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣.∣∣∣∣h′′(η − t η ν + t η ω ) ∣∣∣∣qdt ≤ |h′′(ν)|q ηs ξ2k(µ, α) + |h′′(ω)|q ηs ξ1k(µ, α) (55) Similarly ∫ 1 0 ∣∣∣∣ ∫ t 0 ( 1− (1− φ)µ ku )α k dφ ∣∣∣∣.∣∣∣∣h′′( t η ν + η − t η ω ) ∣∣∣∣qdt ≤ |h′′(ν)|q ηs ξ1k(µ, α) + |h′′(ω)|q ηs ξ2k(µ, α). (56) Substituting (55) and (56) in equation (51), we obtain required result. Remark 14. By assuming the values of η = 2,s=1 and k = 1 in (51), the resulting outcome is obtained as follow∣∣∣∣ 2µα−1 (ω − ν)µα µαΓ (α+ 1) ( αjµν+ω η −h(ν) + α jµν+ω η + h(ω) ) − h ( ν + ω 2 ) ∣∣∣∣≤ (ω − ν)2 8 µα × ( ξ1(µ, α) )1− 1 q [( 2ξ1(µ, α)− ξ2(µ, α) 2 |h′′(ν)|q + ξ2(µ, α) 2 |h′′(ω)|q ) 1 q + ( ξ2(µ, α) 2 |h′′(ν)|q + 2ξ1(µ, α)− ξ2(µ, α) 2 |h′′(ω)|q ) 1 q ] . (57) M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 29 of 34 Remark 15. Substituting α = 1 and µ = 1 in (57), we get the following inequality∣∣∣∣ 1 (ω − ν) ∫ ω ν h(x)dx− h′ ( ν + ω 2 ) ∣∣∣∣≤ (ω − ν)2 8 ( 1 6 )1− 1 q [( 5 8 |h′′(ν)|q + 3 8 |h′′(ω)|q ) 1 q + ( 3 8 |h′′(ν)|q + 5 8 |h′′(ω)|q ) 1 q ] . (58) Example 6. This example illustrates the use of Theorem 6 through a combination of graphical and numerical techniques. The function under consideration is h(x) = x6+2x4, defined on the interval [2, 7], with the inequality evaluated using the parameters k = 3, α = 4,s=1, 1 p = 0.6, 1 q = 0.4, and η = 8. Explanation: Figure 11 provides a 2D visualization of the inequality (51), depicting the behavior of the left-hand side and right-hand side as µ varies in (0, 1]. The graph demonstrates the validity of the theorem. To complement the graphical analysis, numerical evaluations of the LHS and RHS were conducted for specific values of µ. The results, displayed in a Table 8, confirm that the inequality holds under the given parameters. Figure 11: Verification of Theorem 6 using graphical and numerical data for µ ∈ (0, 1]. Table 8: This table offers numerical data for comparison of left hand side and right hand side of inequality (51). µ 0.2 0.4 0.6 0.8 1 LHS 65.30 144.68 221 291.37 355.37 RHS 115.67 255.29 388.67 510.90 621.48 Further validation was performed by generating a 3D representation of the inequality as α ranges within [5, 10] and µ within (0, 1]. The resulting surface plot, presented in Figure 12, illustrates that the inequality remains valid across the explored parameter space. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 30 of 34 Figure 12: Three-dimensional validation of Theorem 6 across different values of α and µ. This comprehensive analysis underscores the utility and reliability of Theorem 6, veri- fying its capacity to constrain the behavior of h(x) under the specified conditions. 5. Conclusion In this paper, we have successfully extended the understanding of inequalities associ- ated with twice differentiable functions through the lens of extended conformable fractional operators. The establishment of new equalities and the derivation of novel trapezoidal- type and midpoint-type inequalities highlight the significance convexity. The applications of established inequalities, such as the power mean inequality and Hölder’s inequality has led to the development of a new class of inequalities, further enriching the existing body of knowledge. These findings not only generalize previous research but also open avenues for future exploration in the field of fractional calculus. Importantly, this research bridges a gap between classical convex analysis and the modern framework of fractional calculus, emphasizing the role of extended operators in advancing mathematical theory. Furthermore, the insights gained from this study may inspire researchers to investigate the application of these concepts to other fractional operators. This study lays a solid foundation for future investigations, particularly in applying extended conformable frac- tional operators to other forms of inequalities or to different classes of functions beyond the twice differentiable case. This work encourages applying these concepts to other frac- tional operators and highlights the link between convexity, differentiability, and fractional calculus. Acknowledgements Authors M.Sarwar, N.Fatima and K. Abodayeh are thankful to Prince Sultan Univer- sity for APC and support through TAS research lab. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 31 of 34 Declarations: Availability of data and material The data used to support the findings of this study are available from the corresponding author upon request. Funding This work does not receive any external funding. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final version. Competing interests The authors declare that they have no conflicts of interest. References [1] Ohta and Shin-ichi. Uniform convexity and smoothness, and their applications in finsler geometry. Mathematische Annalen, 343:669–699, 2009. [2] Mecke and Klaus R. Additivity, convexity, and beyond: applications of minkowski functionals in statistical physics. statistical physics and spatial statistics: The art of analyzing and modeling spatial structures and pattern formation. Berlin, Heidelberg: Springer Berlin Heidelberg, pages 111–184, 2000. [3] Michael Hirsch and Wolfgang Quapp. Reaction pathways and convexity of the po- tential energy surface: application of newton trajectories. Journal of mathematical chemistry, 36:307–340, 2004. [4] El-Shahed and Moustafa. Fractional calculus model of the semilunar heart valve vibrations. International design engineering technical conferences and computers and information in engineering conference., 37033, 2003. [5] Alberto Cambini and Laura Martein. Generalized convexity and optimization: The- ory and applications. Springer Science and Business Media., 616, 2008. [6] Magin and R L. Fractional calculus in bioengineering begell house publishers. Inc., Connecticut., 2006. [7] Pham Dinh Tao and LT Hoai An. Convex analysis approach to dc programming: theory, algorithms and applications. Acta mathematica vietnamica., 22(1):289–355, 1997. [8] Xinyue Shen and et al. Disciplined multi-convex programming. 29th Chinese control and decision conference (CCDC) IEEE., 2017. [9] Asimow and Leonhard. Convexity theory and its applications in functional analysis. Elsevier., 16, 2014. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 32 of 34 [10] Hasanov and Alemdar. Convexity argument for monotone potential operators and its application. Nonlinear Analysis-Series A Theory and Methods and Series B Real World Applications., 41(7):907–920, 2000. [11] Soĺıs-Pérez J. E. Gomez-Aguilar J. F. Razo-Hernández J. R. Etemad S. Lav́ın- Delgado, J. E. and S. Rezapour. An improved object detection algorithm based on the hessian matrix and conformable derivative. Circuits, Systems, and Signal Processing., 43(8):4991–5047, 2024. [12] Krantz and Steven G. Convexity in complex analysis. Proc. Symp. Pure Math., 52, 1991. [13] Archimedes and Sir Thomas Little Heath. The works of archimedes. CUP Archive, 2004. [14] Kjeldsen and Tinne Hoff. From measuring tool to geometrical object: Minkowski’s development of the concept of convex bodies. Archive for history of exact sciences., 62(1):59–89, 2008. [15] Steele and J. Michael. The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities. Cambridge University Press, 2004. [16] Smallwood and Peter D. An introduction to risk sensitivity: The use of jensen’s inequality to clarify evolutionary arguments of adaptation and constraint. American Zoologist., 36(4):392–401, 1996. [17] Miao-Kun Wang and et al. An optimal power mean inequality for the complete elliptic integrals. Applied Mathematics Letters., 24(6):887–890, 2011. [18] Roger A. Horn and Roy Mathias. Cauchy-schwarz inequalities associated with positive semidefinite matrices. Linear Algebra and its Applications., 142:63–82, 1990. [19] Khrennikov and A. Yu. Epr-bohm experiment and bell’s inequality: Quantum physics meets probability theory. Theoretical and Mathematical Physics., 157:1448–1460, 2008. [20] Karl Hess De Raedt, Hans and Kristel Michielsen. Extended boole-bell inequalities applicable to quantum theory. Journal of computational and theoretical nanoscience., 8(6):1011–1039, 2011. [21] Wu and Liming. A new modified logarithmic sobolev inequality for poisson point processes and several applications. Probability Theory and Related Fields., 118(3):427– 438, 2000. [22] GR Mohtashami Borzadaran and D. N. Shanbhag. Further results based on chernoff- type inequalities. Statistics and probability letters., 39(2):109–117, 1998. [23] Sever S. Dragomir and Charles EM Pearce. Selected topics on Hermite-Hadamard inequalities and applications. SSRN, Melbourne City MC, Victoria, Australia, 2018. [24] Muhammad Samraiz Cetin Yıldız Maryam Ali Alghafli Rahman, Gauhar and Nabil Mlaiki. Advancements in ostrowski type fractional integral inequalities via applica- tions of jensen’s and young’s inequalities. European Journal of Pure and Applied Mathematics, 18(2):5816–5816, 2025. [25] Kirmaci and Ugur S. Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Applied mathematics and computation., 147(1):137–146, 2004. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 33 of 34 [26] S S Agarwal Dragomir and R. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Applied mathematics letters., 11(5):91–95, 1998. [27] Rajpar A. H. Alzabut J. Aslam M.-Etemad S. Rezapour Khan, H. and S. On a fractal–fractional-based modeling for influenza and its analytical results. Qualitative theory of dynamical systems., 23(2):70, 2024. [28] Lav́ın-Delgado J. E. Gómez-Aguilar-J. F. Razo-Hernández-J. R. Etemad S. Rezapour Chávez-Vázquez, S. and S. Trajectory tracking of stanford robot manipulator by fractional-order sliding mode control. Applied Mathematical Modelling., 120:436–462, 2023. [29] Hari M Srivastava Kilbas, Anatolĭı Aleksandrovich and Juan J Trujillo. Theory and applications of fractional differential equations. elsevier, 2006. [30] Soubhagya Kumar et al. Sahoo. New fractional integral inequalities for convex func- tions pertaining to caputo–fabrizio operator. Fractal and Fractional., 6(3):171, 2022. [31] Rudolf Hilfer and ed. Applications of fractional calculus in physics. World scientific., Shelton street, Covent Garden, London., 2000. [32] Kilbas and Anatoly A. Hadamard-type fractional calculus. Journal of the Korean Mathematical Society., 38(6):1191–1204, 2001. [33] Katugampola and Udita N. A new approach to generalized fractional derivatives. arXiv preprint arXiv., 1106.0965, 2011. [34] Jarad and Fahd et al. On a new class of fractional operators. Advances in Difference Equations., pages 1–16, 2017. [35] Samraiz M. Haque S. Rahman, G. and N. Mlaiki. Exploration of some novel integral inequalities pertaining to the new class of (k, ρ)−conformable fractional integrals. Contemporary Mathematics., pages 2853–2877, 2025. [36] Samraiz M Rahman G-Haque S Aloqaily A-Mlaiki N. Younis M, Mehmood A. Com- putational representation of fractional inequalities through 2d and 3d graphs with applications. Computation., 13(2):46, 2025. [37] Josip Pecaric Mitrinovic, Dragoslav S. and Arlington M. Fink. Classical and new inequalities in analysis. Springer Science and Business Media., 61, 2013. [38] Srivastava H.M. Trujillo Kilbas, A. and J.J. Theory and application of fractional differential equations. North Holland: Elsevier., 2006. [39] Habib S. Mubeen S. Qi, F. and Naeem M. N. Generalized k-fractional conformable integrals and related inequalities. AIMS Mathematics., 4(3):343–358, 2019. [40] Kenneth S. Miller and Bertram Ross. An introduction to the fractional calculus and fractional differential equations. John Wiley and Sons., 1993. [41] Miller S. Ross and B. An introduction to the fractional calculus and fractional dif- ferential equations. John Wiley and Sons., 1993. [42] Sever S. Dragomir and Charles EM Pearce. Selected topics on hermite-hadamard inequalities and applications. SSRN., 2018. [43] Kavurmaci H. Avci, M., Özdemir, and M.E. New inequalities of hermite–hadamard type via s-convex functions in the second sense with applications. Applied Mathemat- ics and Computation., 217(12):5171–5176, 2011. M. Samraiz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6139 34 of 34 [44] Budak H. Etemad S. Rezapour S.-Ahmad H. Kaabar Kara, H. and M. K. A study on the new class of inequalities of midpoint-type and trapezoidal-type based on twice differentiable functions with conformable operators. Journal of Function Spaces., page 4624604, 2023(1). [45] Pshtiwan Othman Mohammed and Mehmet Zeki Sarikaya. On generalized fractional integral inequalities for twice differentiable convex functions. Journal of Computa- tional and Applied Mathematics., 372:112740, 2020. [46] Mehmet Zeki Sarikaya and Nesip Aktan. On the generalization of some integral inequalities and their applications. Mathematical and computer Modelling., 54.9- 10:2175–2182, 2011.