Int. J. Anal. Appl. (2022), 20:8 n-Convexity via Delta-Integral Representation of Divided Difference on Time Scales Hira Ashraf Baig∗, Naveed Ahmad School of Mathematics and Computer Sciences, Institute of Business Administration, Karachi, Pakistan ∗Corresponding author: habaig@iba.edu.pk Abstract. We introduce the delta-integral representation of divided difference on arbitrary time scales and utilize it to set criteria for n-convex functions involving delta-derivative on time scales. Conse- quences of the theory appear in terms of estimates which generalize and extend some important facts in mathematical analysis. 1. Introduction Time scale calculus is a well known and rapidly growing theory in mathematical analysis which unifies two distinct well-known mathematical areas named as continuous and discrete analysis. For supplementary details and basics of time scale calculus, we invoke [1–3]. The notion of convexity with its various types have a noteworthy presence in literature, see [4–7] and the references therein. The notion is firstly generalized on an arbitrary time scale in 2008 by Cristian Dinu [8], subsequently a large number of estimation and inequalities for the functions that are convex on time scales are in the continuous state of development, some of them are present in [9,10]. Here we consult with an exclusive variety of these functions, that is n-convex functions. The n-convexity or higher order convexity firstly investigated by Eberhard Hopf [11] in his scholarly thesis. Further it was discussed in different narrations by Popoviciu [12,13]. A comprehensive review of this family of functions is elaborated in [5, 14]. In [15] M. Rozarija, and J. Pečarić discussed some "Jensen-Type Inequalities on Time Scales" involving real-valued n-convex functions. Higher order convex functions has been discussed on time scales with constant graininess function by H. A. Baig and N. Ahmad in [16], so there is a need to explore this class of functions on arbitrary time scales. Received: Nov. 18, 2021. 2010 Mathematics Subject Classification. 26D20, 39B62, 36A51, 34N05. Key words and phrases. n-convex functions; delta integrals; Time scales; integral inequalities. https://doi.org/10.28924/2291-8639-20-2022-8 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-8 2 Int. J. Anal. Appl. (2022), 20:8 This article is structured as follows. In section 2 we furnish few preliminaries, utilizing in the main results. Section 3 is dedicated to construct a relationship between nth delta derivatives and nth-order divided difference on arbitrary time scales. Afterward, we presented some mathematical inequalities as consequences of our main results in the last section. 2. Preliminaries A time scale T is defined to be an arbitrary closed subset of the real numbers R, with the standard inherited topology. The forward jump operator and the backward jump operator are defined by σ(t) := inf{s ∈ T : s > t}, and ρ(t) := sup{s ∈ T : s < t}, where infφ = supT and supφ = infT. Let u : T→R, u∆(t) is representing the first delta derivative of function u at t ∈Tκ. The second-order delta derivative of u at t is defined as, provided it exists u∆ 2 (t) = u∆∆(t) = (u∆(t))∆ : Tκ 2 →R Similarly higher-order derivatives are defined as u∆ n (t) : Tκ n → R. The definition for rd-continuous functions can be seen in [2]. The set of rd-continuous functions u : T→R is denoted by Crd = Crd(T,R) = Crd(T). The set consisting of first-order delta differentiable functions u and whose derivative is rd-continuous is denoted by C1rd = C 1 rd(T,R) = C 1 rd(T). The substitution rule and first mean value theorem for delta-integrals in time scales are presented in [1–3]. Theorem 2.1. Assume ν : T→R is strictly increasing and T̃ := ν(T) is a time scale. If u ∈ Crd and ν ∈ C1rd, then for a,b ∈T ∫ b a u(t)ν∆(t)∆t = ∫ ν(b) ν(a) ( u ◦ν−1 ) (s)∆̃s. (2.1) Theorem 2.2. Let ν and u be bounded and integrable functions on [a,b], and let ν be nonnegative (or nonpositive) on [a,b]. Let us set M = sup{u(t) : t ∈ [a,b)} m = inf{u(t) : t ∈ [a,b)}. Then there exists a real number λ satisfying the inequalities m < λ < M such that∫ b a u(t)ν(t)∆t = λ ∫ b a ν(t)∆t. The time scale monomials have been defined in [1,3,17] recursively as g0(t,s) = h0(t,s) = 1 for s,t ∈T, Int. J. Anal. Appl. (2022), 20:8 3 gk+1(t,s) = ∫ t s gk (σ(γ),s) ∆γ, hk+1(t,s) = ∫ t s hk (γ,s) ∆γ, k ∈N0. (2.2) These monomials satisfy the following relation for t ∈T and s ∈Tκ: gn (t,s) = (−1)nhn (s,t) . (2.3) Remark 2.1. [17] The functions hn and gn satisfy gn(t,s) ≥ 0 and hn(t,s) ≥ 0 for all t ≥ s. Let us recall the Taylor’s formula defined on time scales from [17]. Theorem 2.3. Let u be n-times delta-differentiable on Tκ n , t ∈T and tα ∈Tκ n−1 . We have u(t) − n−1∑ k=0 hk(t,tα)u ∆k (tα) = ∫ ρn−1(t) tα hn−1(t,σ(γ))u ∆n (γ)∆γ, (2.4) similarly, u(t) − n−1∑ k=0 (−1)ngk(tα,t)u∆ k (tα) = ∫ ρn−1(t) tα (−1)ngn−1(σ(γ),t)u∆ n (γ)∆γ, (2.5) where k ∈N0. higher order convex functions defined on R as well as on Z through nth-order divided difference, in which we randomly select n + 1 points {a0,a1, . . . ,an} from R or from Z, respectively and compute the nth-order divided difference by the formula [a0,a1, · · · ,an; u] = [a1,a2, · · · ,an; u] − [a0,a1, · · · ,an−1; u] an −a0 . (2.6) If (2.6) is non-negative we say that u is an n-convex function. Here (2.6) remains same for every permutation of n + 1 points. To construct the criteria for n-convexity we need to introduce the forward operator σ in the definition of higher order convexity. So we adopt the same strategy as we did in [16]. Assume n + 1 distinct points t0, · · · ,tn ∈T and arrange them in an increasing order. Relabel these points in the time scale T̃ in terms of forward operator, that is T̃ = {t0,σ(t0), · · · ,σn(t0)}. Consequently we can define the nth-order divided difference for n + 1 points as [t0,σ(t0), · · · ,σn(t0); u] = [σ(t0),σ 2(t0), · · · ,σn(t0); u] − [t0,σ(t0), · · · ,σn−1(t0); u] σn(t0) − t0 . (2.7) So a function u : T→R, is said to be n-convex if [t0,σ(t0), · · · ,σn(t0); u] ≥ 0, (2.8) where σ : T ⋂ T̃→T ⋂ T̃. 4 Int. J. Anal. Appl. (2022), 20:8 3. Main Results Here we want to establish a criteria for n-convex function on arbitrary time scales which is stated as u ∈ Cnrd is n-convex iff u ∆n ≥ 0. It is sufficient to prove this on T̃. Firstly we introduce a new representation of divided difference in terms of delta-integral, that can be seen in the next Theorem. Theorem 3.1. Suppose u ∈ Cnrd(T,R). Let t0,t1, · · · ,tn be n + 1 distinct points in T, then [t0,σ(t0), · · · ,σn(t0); u] = ∫ 1 0 ∆s1 ∫ s1 0 ∆s2 · · · ∫ sn−1 0 ∆sn ×u∆ n (sn[σ n(t0) −σn−1(t0)] + · · · + s1[σ(t0) − t0] + t0), (3.1) where n ≥ 1 and si ∈ [0, 1]. Proof. Consider t0,t1, · · · ,tn, n + 1 distinct points and the corresponding time scale T̃ = {t0,σ(t0), · · · ,σn(t0)}. We prove (4.3) by induction method. For this we first show that [t0,σ(t0); u] = ∫ 1 0 u∆(s1[σ(t0) − t0] + t0)∆s1. (3.2) Let us use the time scales substitution rule for integration (2.1), let the new variable of integration β in the following manner (since σ(t0) 6= t0) β = v−1(s1) = s1[σ(t0) − t0] + t0 ⇒ v(s1) = s1 − t0 σ(t0) − t0 , here v−1 : [0, 1] → T̃. By calculating delta derivative of v(s1) with respect to s1 we get v∆(s1) = 1 σ(t0)−t0 therefore, s1 ∈ [0, 1] and v(s1) is strictly increasing such that v[t0,σ(t0)] = [0, 1]. Hence the corresponding limits are (s1 = 0) → (β = t0); (s1 = 1) → (β = σ(t0)). Since σ(t0) 6= t0, thus (3.2) can be written as∫ 1 0 u∆(s1[σ(t0) − t0] + t0)∆s1 = ∫ v(σ(t0)) v(t0) u∆(v−1(s1))∆s1 = ∫ σ(t0) t0 u∆(β) σ(t0) − t0 ∆β = 1 σ(t0) − t0 ( u(β) ∣∣∣∣σ(t0) t0 ) = u(σ(t0)) −u(t0) σ(t0) − t0 . Int. J. Anal. Appl. (2022), 20:8 5 Now we make the inductive hypothesis that [t0,σ(t0), · · · ,σn−1(t0); u] = ∫ 1 0 ∆s1 ∫ s1 0 ∆s2 · · · ∫ sn−2 0 ∆sn−1 ×u∆ n−1 (sn−1[σ n−1(t0) −σn−2(t0)] + · · · + s1[σ(t0) − t0] + t0). In the integral in (3.1) we apply substitution rule of integration of time scales (2.1) by replacing the variable of integration sn with β. β =v−1(sn) = sn[σ n(t0) −σn−1(t0)] + · · · + s1[σ(t0) − t0] + t0 ⇒ v(sn) = sn − (sn−1[σn−1(t0) −σn−2(t0)] + · · · + s1[σ(t0) − t0] + t0) σn(t0) −σn−1(t0) . So that the delta derivative of v(sn) with respect to sn gives us v∆(sn) = 1 σn(t0) −σn−1(t0) . The corresponding limits are (sn = 0) → ( β = β0 ≡ sn−1[σn−1(t0) −σn−2(t0)] + · · · + s1[σ(t0) − t0] + t0 ) (sn = sn−1) → (β = β1 ≡ sn−1[σn(t0) −σn−2(t0)] + sn−2[σn−2(t0) −σn−3(t0)]+ · · · + s1[σ(t0) − t0] + t0). Thus the innermost integral of (4.3) can transform in the following manner, since σn(t0) 6= σn−1(t0) ∫ sn−1 0 u∆ n (sn[σ n(t0) −σn−1(t0)]) + · · · + s1[σ(t0) − t0] + t0)∆sn = ∫ β1 β0 u∆ n (β) σn(t0) −σn−1(t0) ∆β = 1 σn(t0) −σn−1(t0) ( u∆ n−1 (β) ∣∣∣∣β1 β0 ) = u∆ n−1 (β1) −u∆ n−1 (β0) σn(t0) −σn−1(t0) . However, by applying the inductive hypothesis we have ∫ 1 0 ∆s1 ∫ s1 0 ∆s2 · · · ∫ sn−2 0 ∆sn−1 ( u∆ n−1 (β1) −u∆ n−1 (β0) σn(t0) −σn−1(t0) ) = u[t0,σ(t0), · · · ,σn−2(t0),σn(t0)] −u[t0,σ(t0), · · · ,σn−2(t0),σn−1(t0)] σn(t0) −σn−1(t0) = [t0,σ(t0), · · · ,σn(t0); u]. 6 Int. J. Anal. Appl. (2022), 20:8 � In the next Theorem we establish a relation between nth-order divided difference and nth-delta derivative on arbitrary time scales, since in this result the points ti ∈T need not to be distinct. Theorem 3.2. Let u ∈ Cnrd(T,R), then for n + 1 points form T we have [t0,σ(t0), · · · ,σn(t0); u] = u∆ n (ξ) (hi (sn−i, 0)) , (3.3) where s0 = 1, 0 ≤ i ≤ n, and ξ ∈ [t0,σn(t0)]T. Proof. By using the time scale monomials (2.2) we can write a general notation for the integral∫ 1 0 ∆s1 ∫ s1 0 ∆s2 · · · ∫ sn−1 0 ∆sn, that is hi (sn−i, 0) = ∫ sn−i 0 hi−1(sn−i+1, 0)∆sn−i+1. (3.4) By the Remark 2.1 we can conclude that hn(si, 0) > 0 in (3.4) because all si > 0. Now by applying Theorem 2.2, (3.1) yields x (hi (sn−i, 0)) ≤ [t0,σ(t0), · · · ,σn(t0); u] ≤ X (hi (sn−i, 0)) , or x ≤ [t0,σ(t0), · · · ,σn(t0); u] (hi (sn−i, 0)) ≤ X, where x ≡ min u∆ n (t) and X ≡ max u∆ n (t) for t ∈ [t0,σn(t0)]T. Then by the rd-continuity of u∆ n there exists a λ in this interval that is u∆ n (ξ) = λ, such that [t0,σ(t0), · · · ,σn(t0); u] (hi (sn−i, 0)) = u∆ n (ξ). � Here, we can directly achieve the next result. Corollary 3.1. Let u : T→R is n-convex function iff u∆ n ≥ 0, given that u∆ n exists. Another useful property of n-convex function is represented in the next result. Theorem 3.3. Let u(t) ∈ Cnrd(T,R) is n-convex function, then for every r ∈ N, 1 ≤ r ≤ n − 1, u ∆r is (n− r)-convex. Proof. By Corollary 3.1 u∆ n ≥ 0. Since u∆ r exists for every 1 ≤ r ≤ n− 1. Let us choose (n− r + 1) points from [ta,tb]T such that T̃ = {t0,σ(t0), · · ·σn−r (t0)}, then by using (3.3) we can write [t0,σ(t0), · · · ,σn−r (t0); u∆ r ] = ( u∆ r (ξ) )∆n−r (hn−r (sr, 0)) = (u(ξ)) ∆n (hn−r (sr, 0)) ≥ 0, (3.5) where ξ ∈ [t0,σn−r (t0)]T. Thus (3.5) shows that u∆ r is (n− r)-convex for every 1 ≤ r ≤ n− 1. � Int. J. Anal. Appl. (2022), 20:8 7 4. Applications: Inequalities for n-convex functions Let us present Levinson’s type inequality for higher-order convex functions on time scales for this we require the next result. Let ti ∈ [ta,tb]T, for i = 1, · · · ,z. Let bi > 0 such that ∑z i=1 bi = 1 therefore t ∈ [ta,tb]T denoted by ∑z i=1 biti. Theorem 4.1. Let u is (n + 2)-convex on T. Then for every t ∈T the function U(t) = [t,σ(t), · · · ,σn(t); u], (4.1) is a convex function. Proof. By using (3.1), (4.1) can be expressed as U(t) = [t,σ(t), · · · ,σn(t); u] = ∫ 1 0 ∫ s1 0 · · · ∫ sn−1 0 u∆ n (sn[σ n(t) −σn−1(t)] + · · · + s1[σ(t) − t] + t)∆sn · · ·∆s1. Therefore u∆ n is convex by Theorem 3.3, thus for fixed sj, σj(t) for j = 1, · · · ,n we can write u∆ n   n∑ j=1 sj[σ j(t) −σj−1(t)] + z∑ i=1 biti   ≤ z∑ i=1 biu ∆n   n∑ j=1 sj[σ j(t) −σj−1(t)] + ti   , which concludes the proof. � Theorem 4.2. If u is (n + 2)-convex on T, then the given inequality is true u[t,σ(t), · · · ,σn(t)] ≤ z∑ i=1 bi [ti,σ(ti ), · · · ,σn(ti ); u]. (4.2) Proof. The proof is the direct consequence of Theorem 4.1. � Remark 4.1. Let T = R in Theorem 4.2, inequality (4.2) coincides with inequality (4) in [18], this Levinson’s type inequality itself having a great importance in literature which is used to develop further divided difference estimates for n-convex functions in [19]. Further, we present certain useful inequalities involving n-convex functions on time scales by using the criteria for n-convexity, that is u∆ n ≥ 0. Theorem 4.3. Let tα,tβ ∈Tκ n , suppose u ∈ Cn+1 rd (T,R) be (n+ 1)-convex function on [tα,tβ]. Then for each t ∈ (tα,tβ), the following inequalities hold n−1∑ k=0 hk(t,tα)u ∆k (tα) + u ∆n (tα) ∫ ρn−1(t) tα hn−1(t,σ(γ))∆γ ≤ u(t) ≤ n−1∑ k=0 hk(t,tα)u ∆k (tα) + u ∆n (tβ) ∫ ρn−1(t) tα hn−1(t,σ(γ))∆γ, (4.3) 8 Int. J. Anal. Appl. (2022), 20:8 where tα < ρn−1(tβ). If n is odd, then n−1∑ k=0 hk(t,tβ)u ∆k (tβ) + u ∆n (tβ) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ ≤ u(t) ≤ n−1∑ k=0 hk(t,tβ)u ∆k (tβ) + u ∆n (tα) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ, (4.4) and if n is even, the given inequality holds n−1∑ k=0 hk(t,tβ)u ∆k (tβ) + u ∆n (tα) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ ≤ u(t) ≤ n−1∑ k=0 hk(t,tβ)u ∆k (tβ) + u ∆n (tβ) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ. (4.5) Proof. If u is (n + 1)−convex on Tκ n which implies that u∆ n+1 ≥ 0, then u∆ n is increasing on Tκ n , i.e u∆ n (tα) ≤ u∆ n (γ) ≤ u∆ n (tβ) for each γ ∈ [tα,tβ], let σ(γ) ≤ t so that hn−1(t,σ(γ)) is non-negative, then from (2.4) we get ∫ ρn−1(t) tα hn−1(t,σ(γ))u(tα)∆γ ≤ u(t) − n−1∑ k=0 hk(t,tα)u ∆k (tα) ≤ ∫ ρn−1(t) tα hn−1(t,σ(γ))u ∆n (tβ)∆γ, which executes the proof for (4.3). Let n is odd and t ≤ σ(γ) so that gn−1(σ(γ),t) ≥ 0, thus we can write∫ tβ ρn−1(t) (−1)n−1gn−1(σ(γ),t)u∆ n (tα)∆γ ≤ ∫ tβ ρn−1(t) (−1)n−1gn−1(σ(γ,t))u∆ n (γ)∆γ ≤ ∫ tβ ρn−1(t) (−1)n−1gn−1(σ(γ),t)u∆ n (tβ)∆γ, ⇒ u∆ n (tβ) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ ≤ ∫ ρn−1(t) tβ hn−1(t,σ(γ))u ∆n (γ)∆γ ≤ u∆ n (tα) ∫ ρn−1(t) tβ hn−1(σ(γ),t)∆γ, Int. J. Anal. Appl. (2022), 20:8 9 which gets the form u∆ n (tβ) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ ≤ u(t) − n−1∑ k=0 hk(t,tβ)u ∆k (tβ) ≤ u∆ n (tα) ∫ ρn−1(t) tβ hn−1(σ(γ),t)∆γ, which executes the proof for (4.4). Let n is even then we have (−1)n−1u∆ n (tβ) ≤ (−1)n−1u∆ n (γ) ≤ (−1)n−1u∆ n (tα), then by adopting the same steps we can prove (4.5). � Therefore, we can extract the particular cases of Theorem 4.3 by considering different time scales. First by taking T = R we obtained the following result which agrees Theorem 1 in [20]. Theorem 4.4. Let u(t) be (n+1)−convex on [tα,tβ]. Then for all t ∈ (tα,tβ), the following inequality holds n∑ k=0 u(k)(tα) k! (t − tα)k ≤ u(t) ≤ n−1∑ k=0 u(k)(tα) k! (t − tα)k + u(n)(tβ) n! (t − tα)n. (4.6) For odd n the following inequality is true n∑ k=0 u(k)(tβ) k! (t − tβ)k ≤ u(t) ≤ n−1∑ k=0 u(k)(tβ) k! (t − tβ)k + u(n)(tα) n! (t − tβ)n, (4.7) and for even n the following inequality holds n−1∑ k=0 u(k)(tβ) k! (t − tβ)k + u(n)(tα) n! (t − tβ)n ≤ u(t) ≤ n∑ k=0 u(k)(tβ) k! (t − tβ)k. (4.8) Now by considering T = Z in Theorem 4.3 we get the discrete analogues of the inequalities (4.6), (4.7) and (4.8). Therefore, σ(t) = t + 1, σn(t) = t + n, ρ(t) = t − 1 and ρn(t) = t −n. Theorem 4.5. Let ut : [tα,tβ] → R be an (n + 1)−convex sequence. Then for all t ∈ (tα,tβ), the following inequality holds n−1∑ k=0 ∆kutα k! (t − tα)k + ∆nutα t−n∑ γ=tα (t −γ − 1)(n−1) (n− 1)! ≤ ut (4.9) ≤ n−1∑ k=0 ∆kutα k! (t − tα)k + ∆nutβ t−n∑ γ=tα (t −γ − 1)(n−1) (n− 1)! . (4.10) 10 Int. J. Anal. Appl. (2022), 20:8 For odd n the following inequality is true n−1∑ k=0 ∆kutβ k! (t − tβ)k + ∆nutβ t−n∑ γ=tβ (t −γ − 1)(n−1) (n− 1)! ≤ ut (4.11) ≤ n−1∑ k=0 ∆kutβ k! (t − tβ)k + ∆nutα t−n∑ γ=tβ (t −γ − 1)(n−1) (n− 1)! , (4.12) and for even n the following inequality holds n−1∑ k=0 ∆kutβ k! (t − tβ)k + ∆nutα t−n∑ γ=tβ (t −γ − 1)(n−1) (n− 1)! ≤ ut (4.13) ≤ n−1∑ k=0 ∆kutβ k! (t − tβ)k + ∆nutβ t−n∑ γ=tβ (t −γ − 1)(n−1) (n− 1)! . (4.14) The next result is obtained by considering n = 1 in (4.3) and (4.4). Corollary 4.1. Let tα,tβ ∈ Tκ, if u is convex on [tα,tβ], then the given inequalities hold for all t ∈ [tα,tβ] max{u(tα) + u∆(tα)(t − tα),u(tβ) + u∆(tβ)(t − tβ)}≤ u(t) ≤ min{u(tα) + u∆(tβ)(t − tα),u(tβ) + u∆(tα)(t − tβ)}. (4.15) The next result is obtained by considering n = 2 in (4.3) and (4.5). Corollary 4.2. Let tα,tβ ∈ Tκ 2 , if u is 3−convex on [tα,tβ], then the given inequalities hold for all t ∈ [tα,tβ]T max { u(tα) + u ∆(tα)(t − tα) + u∆ 2 (tα) ∫ ρ(t) tα (γ − tα)∆γ,u(tβ) + u∆(tβ)(t − tβ) + u∆ 2 (tα) ∫ ρ(t) tα (γ − tβ)∆γ } ≤ u(t) ≤ min { u(tα) + u ∆(tβ)(t − tα) + u∆ 2 (tα) ∫ ρ(t) tα (γ − tβ)∆γ,u(tβ) + u∆(tα)(t − tβ) + u∆ 2 (tβ) ∫ ρ(t) tα (γ − tβ)∆γ } . Remark 4.2. When we take T = R in Corollaries 4.1 and 4.2 we get the results which coincide with Corollary 1 and Corollary 2 in [20] respectively. Moreover Corollary 4.1 for T = R is used to derive more useful result in [21]. 5. Conclusion The notion of n-convexity has been discussed in [16], on specific time scales that are R or hZ. Here we extend the theory on arbitrary time scale and developed the relationship between the delta derivatives of order n and the nth-order divided difference using integral representation of nth-order divided difference on time scales, see [5, 22]. Further we utilized this relationship to derive some Int. 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