International Journal of Analysis and Applications Volume 19, Number 3 (2021), 296-318 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-296 GENERALIZED PETROVIĆ’S INEQUALITIES FOR COORDINATED EXPONENTIALLY m-CONVEX FUNCTIONS WASIM IQBAL1, MUHAMMAD ASLAM NOOR1,∗, KHALIDA INAYAT NOOR1, FARHAT SAFDAR2 1COMSATS University Islamabad, Islamabad, Pakistan 2Department of Mathematics, SBK Women University, Quetta, Pakistan ∗Corresponding author: noormaslam@gmail.com Abstract. In this paper, we introduce a new class of convex function, which is called coordinated expo- nentially m-convex functions. Some new Petrović’s type inequalities for exponentially m-convex functions and coordinated exponentially m-convex functions are derived. Lagrange-type and Cauchy-type mean value theorems for exponentially m−convex and coordinated exponentially m-convex functions are also derived. Several special cases are discussed. We also prove the Lagrange type and Cauchy type mean value theo- rems for exponentially m-convex and coordinated exponentially m−convex functions. Results proved in this paper may stimulate further research in different areas of pure and applied sciences. 1. Introduction Convex functions and their variant forms are being used to study a wide class of problems which arises in various branches of pure and applied sciences. For recent applications, generalizations and other aspects of convex functions, see [1–5, 11–23, 28, 33, 34] and the references therein. One of the most significant inequality is the Petrović’s inequality [25]. Petrović’s type inequality have been obtained by several authors, see [7, 10, 24–31] and reference therein. In recent years, the convexity theory have been generalized in several directions using novel and innovative Received February 2nd, 2021; accepted March 8th, 2021; published April 1st, 2021. 2010 Mathematics Subject Classification. Primary 26A51; Secondary 26D15, 49J40. Key words and phrases. Petrović’s inequality; exponentially m−convex functions; exponentially m−convex functions on coordinates. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 296 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-296 Int. J. Anal. Appl. 19 (3) (2021) 297 techniques. Toader [34] introduced the concepts of m-convex sets and m-convex functions, which appeared to be an interesting generalization of the convex sets and convex functions. Exponentially convex functions were introduced by Bernstein [6], which have applications in covariance analysis. Avriel [4] investigated this concept by imposing the condition of r-convex functions. It is well known that log- convex functions is closely related to exponentially convex functions, which have important and interesting applications in information theory and machine learning. Motivated and inspired by these applications, Noor et. al. [14] considered exponentially convex functions and explored their basic characterizations. They have shown that the optimality conditions of the differentiable exponentially convex functions can be characterized by variational inequalities, which have appeared an interesting field with applications in various fields of pure and applied sciences. For the applications and other aspects of the variational inequalities, see Noor et al.[12, 21] and references therein. Pal and Wong [23] provided the application of exponentially convex functions in information, optimization, statistical theory and related areas. For other aspects of exponentially convex functions and their generalizations, see [2, 5, 12, 14, 16, 18, 19, 21–23, 33]. It is worth mentioning that the exponentially convex functions and m-convex functions are clearly two different generalizations of the convex functions. It is natural to unify these classes. Motivated by these facts, Rashid et al [33] introduced the exponentially m-convex functions and derived some Hermite-Hadamrd type inequalities. These integral inequalities can be used to obtain the upper and lower bounds for the integrand, which have applications in material sciences and numerical analysis. Petrović’s [25] derives some integral inequalities for convex functions, which are known as Petrović’s type inequalities. For the applications and other aspects of Petrović’s inequalities, see [10, 24–27, 29, 31]. In this paper, we introduce some new concepts of coordinated exponentially m-convex functions. We derive Petrović’s type inequality for exponentially m- convex and coordinated exponentially m-convex functions. The Lagrange and Cauchy mean value results for the exponentially m-convex functions are derived. Several important cases are discussed as applications of the obtained results. We expect that the ideas and techniques of this paper may be staring point for further research in this areas. 2. preliminaries In this section, we recall the basic definitions and concepts of the exponentially convex functions. Definition 2.1. A nonempty set Ω ⊆ R is convex, if σu + (1 −σ)v ∈ Ω, ∀u,v ∈ Ω, σ ∈ [0, 1]. Definition 2.2. A function F : Ω → R is convex, if F(σu + (1 −σ)v) ≤ σF(u) + (1 −σ)F(v), ∀u,v ∈ Ω, σ ∈ [0, 1]. Int. J. Anal. Appl. 19 (3) (2021) 298 Toader [34] introduced m-convex functions as follows: Definition 2.3. The function f : [0,b] → R,b > 0, is said to be m-convex, where m ∈ [0, 1], if we have f (σu + m(1 −σ)v) 6 σf(u) + m(1 −σ)f(v), ∀u,v ∈ [0,b],σ ∈ [0, 1]. Remark 2.1. One can note that the notion of m-convexity reduces to convexity for m = 1. For m = 0, we obtain starshaped functions. Noor et al. [12, 14] introduced exponentially convex function as follows: Definition 2.4. A function F is called exponentially convex on Ω, if eF(u+σ(v−u)) ≤ (1 −σ)eF(u) + σeF(v), ∀u,v ∈ Ω, σ ∈ [0, 1],(2.1) which can be written in the following equivalent form, which is due to Avriel [4]. Definition 2.5. A function F is called exponentially convex function on Ω, if F(u + σ(v −u)) ≤ log[(1 −σ)eF(u) + σeF(v)], ∀u,v ∈ Ω, σ ∈ [0, 1].(2.2) For the applications of the exponentially convex functions in information theory and mathematical pro- gramming, see Antczak [3] and Alirezaei and Mathar [2]. Rashid el al.[33] introduced exponentially m−convex function as follows: Definition 2.6. A function F : Ω → R on an interval of real line is exponentially m-convex, where m ∈ (0, 1], if (2.3) eF(σu+m(1−σ)v) ≤ σeF(u) + m(1 −σ)eF(v),∀u,v ∈ Ω,σ ∈ [0, 1]. From now onwards, we take I1 = [a1,b1] and I2 = [c1,d1] as intervals in R. Dragomir [8] introduced coordinated convex functions as follows: Definition 2.7. “ Let us consider the bidimensional interval ∆ = I1 × I2 in R2 with a1 < b1 and c1 < d1. Also, let F : I1 × I2 → R be a mapping. Define partial mappings as” (2.4) Fv : I1 → R defined by Fv(x) = F(x,v) and “ (2.5) Fu : I2 → R defined by Fu(y) = F(u,y). The function F is called coordinated convex, if the partial mappings defined in (2.7) and (2.8) are convex on I1 and I2 respectively, for all v ∈ I2 and u ∈ I1. Int. J. Anal. Appl. 19 (3) (2021) 299 Definition 2.8. The function F : ∆ → R is convex in ∆, if F(σu + (1 −σ)z1,σv + (1 −σ)w1) ≤ σF(u,v) + (1 −σ)F(z1,w1),(2.6) ∀(u,v), (z1,w1) ∈ ∆,σ ∈ [0, 1]. Farid et al.[9] introduced coordinated m-convex functions as follows: Definition 2.9. Let ∆1 = [0,b] × [0,d] ⊂ [0,∞)2, then a function f : ∆ → R is m−convex on coordinates if the partial mappings (2.7) fv : [0,b] → R defined by fv(x) = f(x,v) and (2.8) fu : [0,d] → R defined by fu(y) = f(u,y) are m− convex on [0,b] and [0,d] respectively for all v ∈ [0,d] and u ∈ [0,b]. We now introduce the concept of exponentially m−convex functions on coordinates, which is the main motivation of this paper. Definition 2.10. Let F : ∆1 → R be a positive mapping. The function F is coordinated exponentially m−convex, if the partial mappings defined in (2.7) and (2.8) are exponentially m−convex on [0,b] and [0,d] respectively, for all v ∈ [0,d] and u ∈ [0,b]. Definition 2.11. A positive mapping F : ∆1 → R is exponentially m−convex in ∆1, if eF(σu+m(1−σ)z1,σv+m(1−σ)w1) ≤ σeF(u,v) + m(1 −σ)eF(z1,w1),(2.9) ∀(u,v), (z1,w1) ∈ ∆1,σ ∈ [0, 1],m ∈ (0, 1]. Lemma 2.1. “ Every exponentially m−convex mapping F : ∆1 → R is coordinated exponentially m−convex, but converse is not true in general.” Proof. Let a positive mapping F : ∆1 → R be an exponentially m−convex in ∆1. Also, let Fu : [0,d] → R defined as Fu(v1) := f(u,v1). Then \eFu(σv1+m(1−σ)w1) = eF(u,σv1+m(1−σ)w1) = eF(σu+m(1−σ)z1,σv1+m(1−σ)w1) ≤ σeF(u,v1) + m(1 −σ)eF(z1,w1) = σeFu(v1) + m(1 −σ)eFz1 (w1), ∀σ ∈ [0, 1],v1,w1 ∈ [0,d], ” Int. J. Anal. Appl. 19 (3) (2021) 300 which shows the exponentially m−convexity of Fu. Similarly, one can show the exponentially m−convexity of Fv. Now, consider the positive mapping F0 : [0, 1]2 → [0,∞) given by eF0(u,v1) = uv1. Clearly F is coordinated exponentially m−convex. But it is not exponentially m−convex on [0, 1]2. Indeed, if (u, 0), (0,w1) ∈ [0, 1]2 and σ ∈ [0, 1]. Then eF(σ(u,0)+m(1−σ)(0,w1)) = eF(σu,m(1−σ)w1) = mσ(1 −σ)uw1 and σeF(u,0) + m(1 −σ)eF(0,w1) = 0. Thus, ∀ σ ∈ (0, 1), u,w1 ∈ (0, 1), one has eF(σ(u,0)+m(1−σ)(0,w1)) > σeF(u,0) + m(1 −σ)eF(0,w1), which shows that F is not exponentially m−convex. � Petrović [25] derived some inequality for convex functions. Theorem 2.1. Let (ui, i = 1, 2, ...,n) be non-negative n-tuples and (pj,j = 1, 2, ...,n) be positive n-tuples such that ∑n j=1 pj ≥ 1, n∑ κ=1 pκuκ ∈ [0,a1] and n∑ κ=1 pκuκ ≥ ul for each l = 1, ...,n. Consider the function F is convex on [0,a1], then n∑ κ=1 pκF(uκ) ≤F ( n∑ κ=1 pκuκ ) + ( n∑ κ=1 pκ − 1 ) F(0).(2.10) “ Rehman et al. [31] gave the Petrović’s inequality on coordinated convex functions.” Theorem 2.2. “ Let (ui, i = 1, 2, ...,n) and (vj,j = 1, 2, ...,n) be non-negative n-tuples and (pk,k = 1, ...,n) and (ql, l = 1, ...,n) be positive n-tuples such that ” “ n∑ κ=1 pκ ≥ 1, 0 6= n∑ κ=1 pκuκ ≥ uj for every j = 1, 2, ...,n, ” and “ n∑ l=1 ql ≥ 1, 0 6= n∑ l=1 qlvl ≥ vj for every i = 1, 2, ...,n. Int. J. Anal. Appl. 19 (3) (2021) 301 ” “Let F : [0,a1) × [0,b1) → R be a convex on coordinates, then ” \ n∑ κ=1 n∑ l=1 pκqlF(uκ,vl) 6F ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) + ( n∑ l=1 ql − 1 ) F ( n∑ κ=1 pκuκ, 0 ) (2.11) + ( n∑ κ=1 pκ − 1 )( F ( 0, n∑ l=1 qlvl ) + ( n∑ l=1 ql − 1 ) F(0, 0) ) .” 3. Main Results “ In this section, we prove an important lemma, which plays a key role for proving our next results. Lemma 3.1. “ Let (ui, i = 1, 2, ...,n) be non-negative n-tuples and (pj,j = 1, 2, ...,n) be positive n-tuples such that ∑n j=1 pj ≥ 1, θ ∈ [0,a1],” \ n∑ κ=1 pκuκ ∈ [0,a1] and n∑ κ=1 pκuκ ≥ ul > mθ for each l = 1, ...,n.” “ Suppose a positive function F : [0,a1] → R is exponentially m−convex. If e F(u) u−mθ is increasing on [0,a1], then” \e F ( n∑ κ=1 pκuκ ) ≥ ( n∑ κ=1 pκuκ −mθ ) n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκe F(uκ).”(3.1) Proof. Since ∑n κ=1 pκuκ ≥ ul > mθ for all l = 1, ...,n and eF(u) u−mθ is increasing on [0,a1], we have \ e F ( n∑ κ=1 pκuκ ) ( n∑ κ=1 pκuκ −mθ ) ≥ eF(uκ) (uκ −mθ) . ” This implies (uκ −mθ)e F ( n∑ κ=1 pκuκ ) ≥ ( n∑ κ=1 pκuκ −mθ ) eF(uκ). Multiplying above inequality by pκ and taking sum for κ = 1, ...,n, one has n∑ κ=1 pκ(uκ −mθ)e F ( n∑ κ=1 pκuκ ) ≥ ( n∑ κ=1 pκuκ −mθ ) n∑ l=1 pκe F(uκ), from which, one has the required result. � “ We now derive the Petrović’s type inequality for exponentially m−convex functions. ” Int. J. Anal. Appl. 19 (3) (2021) 302 Theorem 3.1. “Let (ui, i = 1, 2, ...,n) be non-negative n-tuples and (pj,j = 1, 2, ...,n) be positive n-tuples such that ∑n j=1 pj ≥ 1, θ ∈ [0,a1], ” \ n∑ κ=1 pκuκ ∈ [0,a1] and n∑ κ=1 pκuκ ≥ ul > θ for each l = 1, ...,n.” Let a positive function F : [0,∞) → R be an exponentially m−convex. Then (3.2) n∑ l=1 ple F(uκ) ≤ Ae F ( n∑ κ=1 pκuκ ) + m ( n∑ l=1 pl −A ) eF(θ) , where A =   n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ   . Proof. Let a function F be an exponentially m−convex and ℵ(x) = ef(x) −mef(a) x−ma . We take y > x > ma and x = ma + σ(y −ma), where σ ∈ (0, 1). Then ℵ(x) = ef(σy+m(1−σ)a) −mef(a) σy + m(1 −σ)a−ma) ≤ σef(y) + m(1 −σ)ef(a) −mef(a) σ(y −ma) = ef(y) −mef(a) y −ma . This implies ℵ(x) ≤ℵ(y). Hence ℵ(x) is increasing on [0,a]. As we have proved that, if F is exponentially m−convex, then e f(x)−mef(a) x−ma is increasing for x > mθ. Substituting ef(x) by ef(x) −mef(θ) in Lemma 3.1, one has e F ( n∑ κ=1 pκuκ ) −meF(θ) ≥ ( n∑ κ=1 pκuκ −mθ ) n∑ l=1 pl(ul −mθ) n∑ l=1 pl ( eF(uκ) −meF(θ) ) . This gives us n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ ( e F ( n∑ κ=1 pκuκ ) −meF(θ) ) ≥ n∑ l=1 ple F(uκ) −m n∑ l=1 ple F(θ).” Int. J. Anal. Appl. 19 (3) (2021) 303 This leads to n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ e F ( n∑ κ=1 pκuκ ) ≥ n∑ l=1 ple F(uκ) −m n∑ l=1 ple F(θ) +m   n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ  eF(θ). Finally, we have n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ e F ( n∑ κ=1 pκuκ ) ≥ n∑ l=1 ple F(uκ)− m   n∑ l=1 pl − n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ  eF(θ), which is the required result. � If θ = 0, then Theorem 3.1 reduces to the following new result. It can be considered as Petrović’s type inequality for exponentially m−convex function. Theorem 3.2. Let the conditions given in Theorem 3.1 be satisfied and let a positive function F : [0,∞) → R be an exponentially m−convex. Then (3.3) n∑ l=1 ple F(uκ) ≤ e F ( n∑ κ=1 pκuκ ) + m ( n∑ l=1 pl − 1 ) eF(0). If m = 1, then Theorem 3.1 reduces to the following new result. It can be viewed as a new generalized Petrović’s type inequality for exponentially convex function. Theorem 3.3. Let the conditions given in Theorem 3.1 be satisfied. Also, let a positive function F : [0,∞) → R be an exponentially convex. Then (3.4) n∑ l=1 ple F(uκ) ≤   n∑ l=1 pl(ul −θ) n∑ κ=1 pκuκ −θ  eF ( n∑ κ=1 pκuκ ) +   n∑ l=1 pl −   n∑ l=1 pl(ul −θ) n∑ κ=1 pκuκ −θ    eF(θ). If m = 1 and θ = 0, then Theorem 3.1 reduces to the following new result. It can be considered as Petrović’s type inequality for exponentially convex function. Int. J. Anal. Appl. 19 (3) (2021) 304 Theorem 3.4. Let the conditions given in Theorem 3.1 be satisfied and let a positive function F : [0,∞) → R be an exponentially convex. Then (3.5) n∑ l=1 ple F(uκ) ≤ e F ( n∑ κ=1 pκuκ ) + ( n∑ l=1 pl − 1 ) eF(0). Now, we derive the generalized Petrović’s type inequality for coordinated exponentially m−convex func- tions. Theorem 3.5. “ Let (ui, i = 1, 2, ...,n) and (vj,j = 1, 2, ...,n) be non-negative n-tuples and (pk,k = 1, 2, ...,n) and (ql, l = 1, ...,n) be positive n-tuples such that θ ∈ [0,a1], ∑n κ=1 pκ ≥ 1, ∑n l=1 ql ≥ 1, ” \ n∑ κ=1 pκuκ ∈ [0,a1), 0 6= n∑ κ=1 pκuκ ≥ uj > θ for every j = 1, 2, ...,n” and \ n∑ l=1 qlvl ∈ [0,b1), 0 6= n∑ l=1 qlvl ≥ vi > θ for every i = 1, 2, ...,n.” Let a positive function F : [0,∞)2 → R be coordinated exponentially m−convex function. Then (3.6) n∑ κ=1 n∑ l=1 pκqle F(uj,vl) ≤ A { Be F ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) +m ( n∑ l=1 ql −B ) e F ( n∑ κ=1 pκuκ,θ )} + m ( n∑ κ=1 pκ −A ){ Be f ( θ, n∑ l=1 qlvl ) + m ( n∑ l=1 ql −B ) ef(θ,θ) } , where A =   n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  (3.7) and B =   n∑ l=1 ql(vl −mθ) n∑ l=1 qlvl −mθ   .(3.8) Proof. “ Consider the partial mappings Fu : [0,a1] → R and Fv : [0,b1] → R defined by Fu(v1) = F(u,v) and Fv(u) = F(u,v). As F is coordinated exponentially m−convex on [0,∞)2. Therefore, the partial mapping Fv is exponen- tially m−convex on [0,b1]. By Theorem 3.1, we have Int. J. Anal. Appl. 19 (3) (2021) 305 n∑ κ=1 pκe Fv(uj) ≤   n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  eFv ( n∑ κ=1 pκuκ ) +m   n∑ κ=1 pκ − n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  eFv(θ). “This is equivalent to ” n∑ κ=1 pκe F(uj,v) ≤   n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  eF ( n∑ κ=1 pκuκ,v ) +m   n∑ κ=1 pκ − n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  eF(θ,v). By setting v = vl, we get n∑ κ=1 pκe F(uj,vl) ≤   n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  eF ( n∑ κ=1 pκuκ,vl ) +m   n∑ κ=1 pκ − n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  eF(θ,vl). Multiplying above inequality by ql and taking sum for l = 1, ...,n, one has (3.9) n∑ κ=1 n∑ l=1 pκqle F(uj,vl) ≤   n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ   n∑ l=1 qle F ( n∑ κ=1 pκuκ,vl ) + m   n∑ κ=1 pκ − n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ   n∑ l=1 qle F(θ,vl). Now again by Theorem 3.1, we have n∑ l=1 qle F ( n∑ κ=1 pκuκ,vl ) ≤   n∑ l=1 ql(vl −mθ) n∑ l=1 qlvl −mθ  eF ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) + m   n∑ l=1 ql − n∑ l=1 ql(vl −mθ) n∑ l=1 qlvl −mθ  eF ( n∑ κ=1 pκuκ,θ ) Int. J. Anal. Appl. 19 (3) (2021) 306 and n∑ l=1 qle f(θ,vj) ≤   n∑ l=1 ql(vl −mθ) n∑ l=1 qlvl −mθ  ef ( θ, n∑ l=1 qlvl ) +m   n∑ l=1 ql − n∑ l=1 ql(vl −mθ) n∑ l=1 qlvl −mθ  ef(θ,θ). Putting these values in inequality (3.9) and using the notations given in (3.7) and (3.8), we get the required result. � If m = 1, then Theorem 3.5 reduces to the following new result. Theorem 3.6. “ Let the conditions given in 3.5 be satisfied. Also, let a positive function F : [0,∞)2 → R be coordinated exponentially m−convex function, then (3.10) n∑ κ=1 n∑ l=1 pκqle F(uj,vl) ≤ C { De F ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) + ( n∑ l=1 ql −D ) e F ( n∑ κ=1 pκuκ,θ )} + ( n∑ κ=1 pκ −C ){ De f ( θ, n∑ l=1 qlvl ) + ( n∑ l=1 ql −D ) ef(θ,θ) } , where C =   n∑ κ=1 pκ(uκ −θ) n∑ κ=1 pκuκ −θ   and D =   n∑ l=1 ql(vl −θ) n∑ l=1 qlvl −θ   . If θ = 0, then Theorem 3.5 reduces to the following new result. Theorem 3.7. Let the conditions given in Theorem 3.5 be satisfied. If F : [0,∞)2 → R be coordinated exponentially m−convex, then (3.11) n∑ κ=1 n∑ l=1 pκqle F(uj,vl) ≤ { e F ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) +m ( n∑ l=1 ql − 1 ) e F ( n∑ κ=1 pκuκ,0 )} + m ( n∑ κ=1 pκ − 1 ){ e f ( 0, n∑ l=1 qlvl ) + m ( n∑ l=1 ql − 1 ) ef(0,0) } . Int. J. Anal. Appl. 19 (3) (2021) 307 If θ = 0 and m = 1, then Theorem 3.5 reduces to the following new result. Theorem 3.8. Let the conditions given in Theorem 3.5 be satisfied. If F : [0,∞)2 → R be coordinated exponentially convex, then (3.12) n∑ κ=1 n∑ l=1 pκqle F(uj,vl) ≤ { e F ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) + ( n∑ l=1 ql − 1 ) e F ( n∑ κ=1 pκuκ,0 )} + ( n∑ κ=1 pκ − 1 ){ e f ( 0, n∑ l=1 qlvl ) + ( n∑ l=1 ql − 1 ) ef(0,0) } . By considering non-negative difference of (3.3), we define the following functional. (3.13) P(eF) = e F ( n∑ κ=1 pκuκ ) + m ( n∑ l=1 pl − 1 ) eF(0) − n∑ l=1 ple F(uκ). Also by considering non-negative difference of (3.11), we define the following functional. (3.14) Υ(eF) = { e F ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) +m ( n∑ l=1 ql − 1 ) e F ( n∑ κ=1 pκuκ,0 )} + m ( n∑ κ=1 pκ − 1 ){ e f ( 0, n∑ l=1 qlvl ) + m ( n∑ l=1 ql − 1 ) ef(0,0) } − n∑ κ=1 n∑ l=1 pκqle F(uj,vl). We need the following lemma. Lemma 3.2. Let a positive function F : [0,b1] → R be an exponentially m−convex such that n1 6 (u−ma1)eF(u)F′(u) −eF(u) + meF(a1) u2 − 2ma1u + ma21 6 N1, ∀u ∈ [0,b1]\{a1} and a1 ∈ (0,b1). Let γ1,γ2 : [0,b1] → R be positive functions defined as γ1(u) = log[N1u 2 −eF(u)] and γ2(u) = log[e F(u) −n1u2], then γ1 and γ2 are exponentially m−convex on [0,b1]. Int. J. Anal. Appl. 19 (3) (2021) 308 Proof. Suppose Pγ1 (u) = eγ1(u) −meγ1(a1) u−ma1 = N1u 2 −eF(u) −mN1a12 + meF(a1) u−ma1 = N1(u 2 −ma12) u−ma1 − eF(u) −meF(a1) u−ma1 . By differentiating with respect to u, one has P ′γ1 (u) = N1 (u−ma1)2u− (u2 −ma21) (u−ma1)2 − (u−ma1)eF(u)F′(u) −eF(u) + meF(a1) (u−ma1)2 . Since u2 − 2ma1u + m2a21 −m 2a21 + ma 2 1 = (u−ma1) 2 −m(m− 1)a21 > 0, by the given condition, one has N1(u 2 −ma12u + ma21) ≥ (u−ma1)e F(u)F′(u) −eF(u) + meF(a1). This implies N1 u2 − 2ma1u + ma21 (u−ma1)2 ≥ (u−ma1)eF(u)F(u) −eF(u) + meF(a1) (u−ma1)2 , N1 u2 − 2ma1u + ma21 (u−ma1)2 − (u−ma1)eF(u)F(u) −eF(u) + meF(a1) (u−ma1)2 ≥ 0. This implies P ′γ1 (u) ≥ 0, ∀u ∈ [0,a1) ∪ (a1,b1]. Similarly, one can show that P ′γ2 (u) ≥ 0, ∀u ∈ [0,a1) ∪ (a1,b1]. This implies that Pγ1 and Pγ2 are increasing on u ∈ [0,a1) ∪ (a1,b1] for all a ∈ (0,b1). Hence by (??), γ1(u) and γ2(u) are exponentially m−convex in [0,b1]. � Here we prove the mean value theorems related to functional defined for Petrović’s inequality for expo- nentially m−convex functions. Theorem 3.9. “ Let (u1, ...,un) ∈ [0,b1], and (p1, ...,pn) be positive n-tuples such that ∑n k=1 pkuk ≥ uj for each j = 1, 2, ...,n. Also let φ(u) = log u2.” If a positive exponentially m−convex function F ∈ C1([0,b1]), then there exist γ ∈ (0,b1) such that P(eF) = (γ −ma1)eF(γ)F′(γ) −eF(γ) + meF(a1) (γ2 − 2ma1γ + ma21) P(eφ),(3.15) Int. J. Anal. Appl. 19 (3) (2021) 309 “ provided that P(eφ) is non zero and a ∈ (0,b1).” Proof. “As F ∈ C1([0,b1]), so there exist real numbers n1 and N1 such that “ n1 6 (u−ma1)eF(u)F′(u) −eF(u) + meF(a1) (u2 − 2ma1u + ma21) 6 N1, ∀u ∈ [0,b1] and a1 ∈ (0,b1). Consider the functions γ1 and γ2 defined in Lemma 3.2. As γ1 is exponentially m−convex in [0,b1], so P(eγ1 ) ≥ 0, that is P ( N1u 2 −eF(u) ) ≥ 0, this gives (3.16) N1P(eφ) ≥P(eF). Similarly γ2 is exponentially m−convex [0,b1], therefore one has (3.17) n1P(φ) 6P(eF). By assumption P(eφ) is non zero, combining inequalities (3.16) and (3.17), one has n1 6 P(eF) P(eφ) 6 N1. Hence there exist v ∈ (0,b1) such that P(eF) P(eφ) = (γ −ma1)eF(γ)F′(γ) −eF(γ) + meF(a1) (γ2 − 2ma1γ + ma21) , which is the required result. � If we take m = 1, then Theorem 3.9 reduces to the following result. Theorem 3.10. Let the conditions given in Theorem 3.9 be satisfied. If F ∈ C1([0,b1]) is a positive exponentially convex function, then there exist γ ∈ (0,b1) such that P(eF) = (γ −a1)eF(γ)F′(γ) −eF(γ) + eF(a1) (γ −a1)2 P(eφ),(3.18) provided that P(eφ) is non zero and a ∈ (0,b1). Theorem 3.11. Let the conditions given in Theorem 3.9 be satisfied. Suppose the positive exponentially m−convex functions F1,F2 ∈ C1([0,b1]), then there exist γ ∈ (0,b1) such that P(eF1 ) P(eF2 ) = (γ −ma1)eF1(γ)F′1(γ) −eF1(γ) + meF1(a) (γ −ma1)eF2(γ)F2′(γ) −eF2(γ) + meF2(a) , “provided that the denominators are non-zero and a1 ∈ (0,b1).” Int. J. Anal. Appl. 19 (3) (2021) 310 Proof. Suppose k ∈ C1([0,b1]) be a function defined as k = log (c1e F1 − c2eF2 ), where c1 and c2 are defined as c1 = P(eF2 ), c2 = P(eF1 ). Then using Theorem 3.9 with F = k, one has (γ −ma1)elog(c1e F1(γ)−c2eF2(γ))(log(c1e F1(γ) − c2eF2(γ)))′ − (c1eF1(γ) − c2eF2(γ)) + m(c1e F1(a) − c2eF2(a)) = 0, this gives (γ −ma1)(c1eF1(γ) − c2eF2(γ))′ − c1eF1(γ) + c2eF2(γ) + mc1eF1(a) −mc2eF2(a) = 0, that is (γ −ma1)(c1eF1(γ)F′1(γ) − c2e F2(γ)F′2(γ)) − c1e F1(γ) + c2e F2(γ) + mc1e F1(a) −mc2eF2(a) = 0, this gives (γ −ma1)c1eF1(γ)F′1(γ) − (γ −ma1)c2e F2(γ)F′2(γ) − c1e F1(γ) + c2e F2(γ) + mc1e F1(a) −mc2eF2(a) = 0, which implies c1 {(γ −ma1)f′1(γ) −f1(γ) + mf1(a)}− c2 {(γ −ma1)f ′ 2(γ) + F2(γ) −mF2(a)} = 0 c1 { (γ −ma1)eF1(γ)F′1(γ) −e F1(γ) + meF1(a) } = c2 { (γ −ma1)eF2(γ)F′2(γ) −e F2(γ) + meF2(a) } . This gives c2 c1 = (γ −ma1)eF1(γ)F′1(γ) −eF1(γ) + meF1(a) (γ −ma1)eF2(γ)F′2(γ) −eF2(γ) + meF2(a) . Putting the values of c1 and c2, one has the required result. � If we take m = 1, then Theorem 3.11 reduces to the following result. Int. J. Anal. Appl. 19 (3) (2021) 311 Theorem 3.12. Let the conditions given in Theorem 3.11 be satisfied. Suppose the positive exponentially convex functions F1,F2 ∈ C1([0,b1]), then there exist γ ∈ (0,b1) such that P(eF1 ) P(eF2 ) = (γ −a1)eF1(γ)F′1(γ) −eF1(γ) + eF1(a) (γ −a1)eF2(γ)F2′(γ) −eF2(γ) + eF2(a) , “provided that the denominators are non-zero and a1 ∈ (0,b1).” Here we state an important lemma that is helpful in proving mean value theorems related to the non- negative functional of Petrovic̀’s inequality for coordinated exponentially m−convex functions. Lemma 3.3. Let ∆ = [0,b1]× [0,d1]. Also, let F : ∆ → R be a positive coordinated exponentially m−convex function such that n1 6 (u−ma1)eF(u,v) ∂∂uF(u,v) −e F(u,v) + meF(a1,v) (u2 − 2ma1u + ma21)v2 6 N1 and n2 6 (v −mc1)eF(u,v) ∂∂vF(u,v) −e F(u,v) + meF(u,c1) (v2 − 2ma1v + mc21)u2 6 N2 ∀u ∈ [0,b1]\{a1}, a1 ∈ (0,b1) and v ∈ [0,d1]\{c1}, c ∈ (0,d1). Consider the functions αv : [0,b1] → R, and αu : [0,d1] → R, defined as α(u,v) = log[max{N1,N2}u2v2 −eF(u,v)] and β(u,v) = log[eF(u,v) − min{n1,n2}u2v2]. Then α and β are coordinated exponentially m−convex. Proof. Suppose the partial mappings αv : [0,b1] → R and αu : [0,d1] → R defined as αv(u) := α(u,v) for all u ∈ (0,b1] and αu(v) := α(u,v) for all v ∈ (0,d]. Pαv (u) = eαv(u) −meαv(a1) u−ma1 = eα(u,v) −meα(a1,v) u−ma1 = elog[max{N1,N2}u 2v2−meF(u,v)] −melog[max{N1,N2}a 2 1v 2−eF(a1,v)] u−ma1 = N1u 2v2 −eF(u,v) −mN1a21v2 + meF(a1,v) u−ma1 = N1 (u2 −ma21)v2 u−ma1 − eF(u,v) −meF(a1,v) u−ma1 . Int. J. Anal. Appl. 19 (3) (2021) 312 Differentiating partially with respect to u, one has P ′αv (u) = N1v 2 (u−ma1)2u− (u 2 −ma21) (u−ma1)2 − (u−ma1)eF(u,v) ∂∂uF(u,v) −e F(u,v) + meF(a1,v) (u−ma1)2 = N1v 2 (u 2 − 2ma1u + ma21) (u−ma1)2 − (u−ma1)eF(u,v) ∂∂uF(u,v) −e F(u,v) + meF(a1,v) (u−ma1)2 By the given condition, one has N1 ≥ (u−ma1)eF(u,v) ∂∂uF(u,v) −e F(u,v) + meF(a1,v) (u2 − 2ma1u + ma21)v2 . Since (u2 − 2ma1u + ma21)v 2 > 0. This implies N1 (u2 − 2ma1u + ma21)v2 (u−ma1)2 ≥ (u−ma1)eF(u,v) ∂∂uF(u,v) −e F(u,v) + meF(a1,v) (u−ma1)2 , N1 (u2 − 2ma1u + ma21)v2 (u−ma1)2 − (u−ma1)eF(u,v) ∂∂uF(u,v) −e F(u,v) + meF(a1,v) (u−ma1)2 ≥ 0. This implies P ′αv (u) ≥ 0, ∀u ∈ [0,ma1) ∪ (ma1,b1]. Similarly, one can show that P ′αu(v) ≥ 0, ∀u ∈ [0,mc1) ∪ (mc1,d1]. This ensure that Pαv is increasing on [0,ma1) ∪ (ma1,b1] for all a1 ∈ [0,b1] and Pαu is increasing on [0,mc1) ∪ (mc1,d1] for all c1 ∈ [0,d1]. By (??), α is exponentially m−convex. Hence by Lemma 2.1, α is coordinated exponentially m−convex. Similarly, one can show that β is coordinated exponentially m−convex. � “Here we give mean value theorems related to the functional defined for Petrovic̀’s type inequality for coordinated exponentially m−convex functions.” Int. J. Anal. Appl. 19 (3) (2021) 313 Theorem 3.13. Let (u1, ...,un) ∈ [0,b1], (v1, ...,vn) ∈ [0,d1] be non-negative n-tuples and (q1, ...,qn), (p1, ...,pn) be positive n-tuples such that∑n k=1 pkuk ≥ uj for each j = 1, 2, ...,n. Also let ϕ(u,v) = log (u 2v2). Let a positive coordinated exponentially m−convex function F ∈ C1(∆), then there exist (γ1,ζ1) and (γ2,ζ2) in the interior of ∆, such that Υ(eF) = (γ1 −ma)eF(γ1,ζ1) ∂∂uF(γ1,ζ1) −e F(γ1,ζ1) + meF(a,ζ1) (γ21 − 2maγ1 + ma2)ζ21 Υ(eϕ)(3.19) and Υ(eF) = (γ2 −ma)eF(γ2,ζ2) ∂∂vF(γ2,ζ2) −e F(γ2,ζ2) + meF(a,ζ2) (γ22 − 2maγ2 + ma2)ζ22 Υ(eϕ),(3.20) provided that Υ(eϕ) is non-zero and a ∈ (0,b1). Proof. As F has continuous first order partial derivative in ∆, so there exist real numbers n1,n2,N1 and N2 such that n1 6 (u−ma1)eF(u,v) ∂∂uF(u,v) −e F(u,v) + eF(a,v) (u2 − 2ma1u + ma21)v2 6 N1 and n2 6 (v −ma1)eF(u,v) ∂∂vF(u,v) −e F(u,v) + eF(u,a) (v2 − 2ma1v + ma21)u2 6 N2, ∀u ∈ (0,b1], v ∈ (0,d] and a ∈ (0,b1). Consider the functions α and β defined in Lemma 3.3. As α is coordinated exponentially m−convex, then Υ(eα) ≥ 0, that is Υ ( N1u 2v2 −eF(u,v) ) ≥ 0, this gives (3.21) N1Υ(e ϕ) ≥ Υ(eF). Similarly β is coordinated exponentially m−convex, therefore one has (3.22) n1Υ(e ϕ) 6 Υ(eF). By assumption Υ(eϕ) is non-zero, so combining inequalities (3.21) and (3.22), one has n1 6 Υ(eF) Υ(eϕ) 6 N1. Int. J. Anal. Appl. 19 (3) (2021) 314 Hence there exists (γ1,ζ1) in the interior of ∆, such that Υ(eF) = (γ1 −ma)eF(γ1,ζ1) ∂∂uF(γ1,ζ1) −e F(γ1,ζ1) + meF(a,ζ1) (γ21 − 2maγ1 + ma2)ζ21 Υ(eϕ). Similarly, one can show that Υ(eF) = (γ2 −ma)eF(γ2,ζ2) ∂∂vF(γ2,ζ2) −e F(γ2,ζ2) + meF(a,ζ2) (γ22 − 2maγ2 + ma2)ζ22 Υ(eϕ), which is the required result. � If we take m = 1, then Theorem 3.13 reduces to the following result. Theorem 3.14. Let the conditions given in Theorem 3.13 be satisfied. Also, let a positive coordinated exponentially convex function F ∈ C1(∆), then there exist (γ1,ζ1) and (γ2,ζ2) in the interior of ∆, such that Υ(eF) = (γ1 −a)eF(γ1,ζ1) ∂∂uF(γ1,ζ1) −e F(γ1,ζ1) + eF(a,ζ1) (γ21 − 2aγ1 + a2)ζ21 Υ(eϕ)(3.23) and Υ(eF) = (γ2 −a)eF(γ2,ζ2) ∂∂vF(γ2,ζ2) −e F(γ2,ζ2) + eF(a,ζ2) (γ22 − 2aγ2 + a2)ζ22 Υ(eϕ),(3.24) provided that Υ(eϕ) is non-zero and a ∈ (0,b1). Theorem 3.15. Let the conditions given in Theorem 3.13 be satisfied. Also let the positive coordinated exponentially m−convex functions F1,F2 ∈ C1(∆), “then there exist (γ1,ζ1) and (γ2,ζ2) in the interior of ∆, such that” Υ(eF1 ) Υ(eF2 ) = (γ1 −ma)eF(γ1,ζ1) ∂∂uF(γ1,ζ1) −e F(γ1,ζ1) + meF(a,ζ1) (γ2 −ma)eF(γ2,ζ2) ∂∂uF(γ2,ζ2) −e F(γ2,ζ2) + meF(a,ζ2) and Υ(eF1 ) Υ(eF2 ) = (γ1 −ma)eF(γ1,ζ1) ∂∂vF(γ1,ζ1) −e F(γ1,ζ1) + meF(a,ζ1) (γ2 −ma)eF(γ2,ζ2) ∂∂vF(γ2,ζ2) −e F(γ2,ζ2) + meF(a,ζ2) , “provided that the denominators are non-zero and a ∈ (0,b1).” Proof. Suppose k = log (c1e F1 − c2eF2 ), “where c1 and c2 are defined as ” c1 = Υ(e F2 ), c2 = Υ(e F1 ). Int. J. Anal. Appl. 19 (3) (2021) 315 Using Theorem 3.13 with F = k, one has (γ −ma)elog(c1e F1−c2eF2 )(γ,ζ) ∂ ∂u log(c1e F1 − c2eF2 )(γ,ζ) −elog(c1e F1−c2eF2 )(γ,ζ) + melog (c1e F1−c2eF2 )(a,ζ) = 0, (γ −ma) ∂ ∂u (c1e F1 − c2eF2 )(γ,ζ) − (c1eF1 − c2eF2 )(γ,ζ) + m(c1e F1 − c2eF2 )(a,ζ) = 0, (γ1 −ma)c1eF1(γ1,ζ1) ∂ ∂u F1(γ1,ζ1) − (γ2 −ma)c2eF2(γ2,ζ2) ∂ ∂u F2(γ2,ζ2) − c1eF1(γ1,ζ1) + c2eF2(γ2,ζ2) + mc1eF1(a,ζ1) −mc2eF2(a,ζ2) = 0, c1 { (γ1 −ma)eF1(γ1,ζ1) ∂ ∂u F1(γ1,ζ1) −eF1(γ1,ζ1) + meF1(a,ζ1) } − c2 { (γ2 −ma)eF2(γ2,ζ2) ∂ ∂u F2(γ2,ζ2) −eF2(γ2,ζ2) + meF2(a,ζ2) } = 0, c1 { (γ1 −ma)eF1(γ1,ζ1) ∂ ∂u F1(γ1,ζ1) −eF1(γ1,ζ1) + meF1(a,ζ1) } = c2 { (γ2 −ma)eF2(γ2,ζ2) ∂ ∂u F2(γ2,ζ2) −eF2(γ2,ζ2) + meF2(a,ζ2) } , c1 { (γ1 −ma) ∂ ∂u eF1(v,u) −eF1(v,u) + eF1(a,u) } = c2 { (γ1 −ma) ∂ ∂u eF2(v,u) −eF2(v,u) + meF2(a,u) } , c2 c1 = (γ1 −ma)eF(γ1,ζ1) ∂∂uF(γ1,ζ1) −e F(γ1,ζ1) + meF(a,ζ1) (γ2 −ma)eF(γ2,ζ2) ∂∂uF(γ2,ζ2) −e F(γ2,ζ2) + meF(a,ζ2) . Similarly, one can show that c2 c1 = (γ1 −ma)eF(γ1,ζ1) ∂∂vF(γ1,ζ1) −e F(γ1,ζ1) + meF(a,ζ1) (γ2 −ma)eF(γ2,ζ2) ∂∂vF(γ2,ζ2) −e F(γ2,ζ2) + meF(a,ζ2) . Putting the values of c1 and c2, one has the required result. � If we take m = 1, then Theorem 3.15 reduces to the following result. Theorem 3.16. Let the conditions given in Theorem 3.13 be satisfied. Also let the positive coordinated exponentially convex functions F1,F2 ∈ C1(∆), “then there exist (γ1,ζ1) and (γ2,ζ2) in the interior of ∆, such that” Υ(eF1 ) Υ(eF2 ) = (γ1 −a)eF(γ1,ζ1) ∂∂uF(γ1,ζ1) −e F(γ1,ζ1) + eF(a,ζ1) (γ2 −a)eF(γ2,ζ2) ∂∂uF(γ2,ζ2) −e F(γ2,ζ2) + eF(a,ζ2) and Υ(eF1 ) Υ(eF2 ) = (γ1 −a)eF(γ1,ζ1) ∂∂vF(γ1,ζ1) −e F(γ1,ζ1) + eF(a,ζ1) (γ2 −a)eF(γ2,ζ2) ∂∂vF(γ2,ζ2) −e F(γ2,ζ2) + eF(a,ζ2) , “provided that the denominators are non-zero and a ∈ (0,b1).” Int. J. Anal. Appl. 19 (3) (2021) 316 4. Conclusion We have defined the coordinated exponentially m-convex functions. Petrović’s type inequality for ex- ponentially m-convex and coordinated exponentially m-convex functions have been derived. We obtained Lagrange-type and Cauchy-type mean value theorems for exponentially m−convex and coordinated expo- nentially m-convex functions. Some new special cases are discovered. It is expected the ideas and techniques of this paper may motivate the researchers working in functional analysis, information theory and statistical theory to find some applications. This is a new path for future research. 5. acknowledgments We wish to express our deepest gratitude to our teachers, colleagues, collaborators and friends, who have direct or indirect contributions in the process of this paper. The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research facilities. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] M. 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