SOME NOVEL ESTIMATIONS FOR DIFFERENT KINDS OF CONVEX FUNCTIONS VIA MODIFIED ATANGANA-BALEANU INTEGRAL OPERATORS

SOME NOVEL ESTIMATIONS FOR DIFFERENT KINDS OF CONVEX FUNCTIONS VIA MODIFIED ATANGANA-BALEANU INTEGRAL OPERATORS BARIS¸ C¸ELIK, ERHAN SET, AND AHMET OCAK AKDEMIR Abstract. Fractional analysis has recently been used quite effectively in the field of inequality theory, as in all subjects of mathematics. Different and new fractional operators obtained with the help of variants created in the kernel structures have contributed to the development of fractional analysis and have also brought new directions to other fields. Atangana-Baleanu fractional integral operator has an important place among fractional integral operators in terms of its non-singular and non-local properties. In this study, new integral inequalities have been proved for different types of convex function classes by using the Modified Atangana-Baleanu fractional integral operator obtained from Atangana-Baleanu fractional integral operator, which produces functional solutions to many dynamic real world problems compared to other operators. In the proof stages of the main findings, basic inequalities such as H¨older, Power-mean, Young and Jensen inequalities have been used and many reduced results have been provided. 1. Introduction and Preliminaries Convex functions are special types of functions that play a central role in mathematical analysis and optimization theory. This structure of convexity provides a great advantage, especially in the analysis of solution sets and minimum values. Different types of convex functions-such as strictly convex, strongly convex, and piecewise convex functions-are selected according to the nature of various problems and directly affect the performance of the solution methods. For example, strictly convex functions guarantee a singular minimum, while strongly convex functions enable optimization algorithms to converge faster. These types play a critical role in both theoretical analysis and algorithm design in disciplines such as nonlinear programming, machine learning, economics, control theory, and statistics. In addition, the geometric simplicity provided by convexity makes the behavior of the solution space more predictable and computable, even in high-dimensional problems. For this reason, convex functions are among the indispensable tools not only in theoretical mathematical studies but also in modeling complex systems in the real world. We will start with some basic concepts that will be used to prove main findings as follows. 2010 Mathematics Subject Classification. 26A33, 26D10, 26D15. Key words and phrases. Quasi-convex function, H¨older inequality, power mean inequality, Young inequality, modified Atangana-Baleanu (AB) fractional integral operators. 1 2 BARIS¸ C¸ELIK, ERHAN SET, AND AHMET OCAK AKDEMIR Definition 1.1. [2] The function ℵ: [r, s]→Ris called a quasi-convex if the following inequality ℵ(ρu + (1 −ρ)v)≤max{ℵ(u),ℵ(v)} holds for all u, v ∈[r, s]and ρ∈[0,1]. Definition 1.2. The function ℵ: [r, s]→Ris η-convex if ℵ(ρu + (1 −ρ)v)≤ ℵ(v) + ρη(ℵ(u),ℵ(v)) for all u, v ∈[r, s],ρ∈[0,1], and η:R×R→Rwith η(ℵ(u),ℵ(v)) ≤ ℵ(u)− ℵ(v)for ℵ(u)≥ ℵ(v). Similarly, ℵis η-concave if ℵ(ρu + (1 −ρ)v)≥ ℵ(v) + ρη(ℵ(u),ℵ(v)). Fractional analysis, as an extension of classical calculus, offers a deep perspective in mathematical analysis by generalizing the concepts of derivative and integral to noninteger degrees. This field has attracted the attention of various scientists especially since the 17th century, but has started to be researched more intensively in recent years with the developing calculation methods and increasing application areas. Fractional derivatives and integrals, unlike classical derivatives, include not only point changes but also the effects of the past behavior of the system. Thanks to this memory structure, fractional analysis provides a great advantage in modeling the effects accumulated over time in fields such as physics, biology, economics and engineering. Fractional models offer more accurate and flexible solutions, especially in phenomena that cannot be adequately explained by classical methods such as anomalous diffusion, viscoelasticity, electrical circuits, control systems and biological tissue modeling. Mathematically, fractional analysis adds a new dimension to the theory of differential equations and defines more general and comprehensive solution spaces beyond classical solutions. In this respect, it provides a strong theoretical basis for more realistic modeling, control and simulation of systems. As a result, fractional calculus is of critical importance in contemporary science and technology, not only in terms of theoretical mathematics but also in understanding and managing complex systems. Some operators that are among the basic concepts of fractional analysis and have become famous for their applications are defined as follows. Definition 1.3. [3,4] The AB-fractional integral operator for ℵ ∈ L1[r, T ]and 0<ρ<1 is defined by AB rIρ ξℵ(ξ) = 1−ρ M(ρ)ℵ(ξ) + ρ M(ρ)Γ(ρ)Zξ r (ξ−η)ρ−1ℵ(ξ)dη. Definition 1.4. [1] The modified left-sided fractional integral operator for ℵ ∈ L1[r, T ] and 0<ρ<1with respect to Ψis defined by MAB rIρ Ψ(ξ)ℵ(ξ) = 1−ρ M(ρ)ℵ(ξ) + ρ M(ρ)Γ(ρ)Zξ r (Ψ(ξ)−Ψ(η))ρ−1Ψ′(η)ℵ(η)dη. SOME NOVEL INTEGRAL INEQUALITIES 3 Definition 1.5. [1] The modified right-sided fractional integral operator for ℵ ∈ L1[r, s] and 0<ρ<1with respect to Ψis defined by MAB sIρ Ψ(ξ)ℵ(ξ) = 1−ρ M(ρ)ℵ(ξ) + ρ M(ρ)Γ(ρ)Zs ξ (Ψ(η)−Ψ(ξ))ρ−1Ψ′(η)ℵ(η)dη. Remark 1.1. If we consider Ψ(ξ) = ξin Definition 1.4, then we get the AB-operator defined in Definition 1.3. 2. Main Results In the main findings part, first of all, an integral identity is reminded. Then, new integral inequalities are obtained by using this integral identity, some convex function types and some basic inequality derivation methods. Since the results contain modified Atangana-Baleanu (AB) fractional integral operators, new integral inequalities, whose proofs are given, introduce new approaches and add a new dimension to the literature. In [5], Rahman et al. established the following identity involving modified AtanganaBaleanu (AB) fractional integral operators: Lemma 2.1. Assume that Ψ:[r, s]→Ris a strictly increasing and positive function with a continuous derivative on [r, s]. Let ℵ: [r, s]→Rbe a differentiable function on (r, s), where ℵ′∈L1[r, s]and r < s. For modified Atangana-Baleanu (AB) fractional integral operators, the following identity holds: (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i =(t−r)ϱ+1 s−rZ1 0 ρϱℵ′(ρt + (1 −ρ)r)dρ −(s−t)ϱ+1 s−rZ1 0 ρϱℵ′(ρt + (1 −ρ)s)dρ, where ϱ∈(0,1],t∈[r, s], and ρ∈[0,1]. In this section, we use the modified Atangana-Baleanu (AB) fractional integral operators to obtain some fractional integral inequalities for the quasi-convex function and η-convex function, respectively, as follows. Theorem 2.1. Assume that Ψ:[r, s]→Ris a strictly increasing and positive function with a continuous derivative on [r, s]. Let ℵ: [r, s]→Rbe a differentiable function on (r, s),ℵ′∈L1[r, s], and r < s. If |ℵ′|qis a quasi-convex function, the following inequality holds: (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤1 ϱp + 1 1 p(t−r)ϱ+1 s−rmax |ℵ′(t)|q,|ℵ′(r)|q 1 q 4 BARIS¸ C¸ELIK, ERHAN SET, AND AHMET OCAK AKDEMIR +(s−t)ϱ+1 s−rmax |ℵ′(t)|q,|ℵ′(r)|q 1 q, where 1 p+1 q= 1,t∈[r, s],ϱ∈(0,1], and M(ϱ)>0. Proof. Using Lemma 2.1, well known H¨older inequality and the quasi-convexity of |ℵ′|q on [r, s], we can write (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−rZ1 0 ρϱp dρ 1 pZ1 0 |ℵ′(ρt + (1 −ρ)r)|qdρ 1 q +(s−t)ϱ+1 s−rZ1 0 ρϱp dρ 1 pZ1 0 |ℵ′(ρt + (1 −ρ)s)|qdρ 1 q ≤(t−r)ϱ+1 s−rZ1 0 ρϱp dρ 1 p max |ℵ′(t)|q,|ℵ′(r)|q 1 q +(s−t)ϱ+1 s−rZ1 0 ρϱp dρ 1 p max |ℵ′(t)|q,|ℵ′(s)|q 1 q =(t−r)ϱ+1 s−r1 ϱp + 1 1 pmax |ℵ′(t)|q,|ℵ′(r)|q 1 q +(s−t)ϱ+1 s−r1 ϱp + 1 1 pmax |ℵ′(t)|q,|ℵ′(s)|q 1 q =1 ϱp + 1 1 p(t−r)ϱ+1 s−rmax |ℵ′(t)|q,|ℵ′(r)|q 1 q +(s−t)ϱ+1 s−rmax |ℵ′(t)|q,|ℵ′(r)|q 1 q where it is easily seen that R1 0ρϱp dρ =1 ϱp+1 . This completes the proof. □ Theorem 2.2. Assume that Ψ:[r, s]→Ris a strictly increasing and positive function with a continuous derivative on [r, s]. Let ℵ: [r, s]→Rbe a differentiable function on (r, s),ℵ′∈L1[r, s], and r < s. If |ℵ′|qis a quasi-convex function, the following inequality holds: (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤1 ϱ+ 1(t−r)ϱ+1 s−rmax |ℵ′(t)|q,|ℵ′(r)|q 1 q+(s−t)ϱ+1 s−rmax |ℵ′(t)|q,|ℵ′(s)|q 1 q, SOME NOVEL INTEGRAL INEQUALITIES 5 where q≥1,t∈[r, s],ϱ∈(0,1], and M(ϱ)>0. Proof. Using Lemma 2.1, well known power mean inequality and the quasi-convexity of |ℵ′|qon [r, s], we can write (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−rZ1 0 ρϱdρ1−1 qZ1 0 ρϱ|ℵ′(ρt + (1 −ρ)r)|qdρ 1 q +(s−t)ϱ+1 s−rZ1 0 ρϱdρ1−1 qZ1 0 ρϱ|ℵ′(ρt + (1 −ρ)s)|qdρ 1 q ≤(t−r)ϱ+1 s−rZ1 0 ρϱdρ1−1 qZ1 0 ρϱmax |ℵ′(t)|q,|ℵ′(r)|qdρ 1 q +(s−t)ϱ+1 s−rZ1 0 ρϱdρ1−1 qZ1 0 ρϱmax |ℵ′(t)|q,|ℵ′(s)|qdρ 1 q =1 ϱ+ 11−1 q1 ϱ+ 1 1 q ×(t−r)ϱ+1 s−rmax |ℵ′(t)|q,|ℵ′(r)|q 1 q+(s−t)ϱ+1 s−rmax |ℵ′(t)|q,|ℵ′(s)|q 1 q. So the proof is completed. □ Theorem 2.3. Assume that Ψ:[r, s]→Ris a strictly increasing and positive function with a continuous derivative on [r, s]. Let ℵ: [r, s]→Rbe a differentiable function on (r, s),ℵ′∈L1[r, s], and r < s. If |ℵ′|qis a quasi-convex function, the following inequality holds: (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−r1 p(ϱp + 1) +1 qmax |ℵ′(t)|q,|ℵ′(r)|q +(s−t)ϱ+1 s−r1 p(ϱp + 1) +1 qmax |ℵ′(t)|q,|ℵ′(s)|q, where 1 p+1 q= 1,t∈[r, s],ϱ∈(0,1], and M(ϱ)>0. Proof. From Lemma 2.1, apply the Young inequality ab ≤ap p+bq q, we obtain (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] 6 BARIS¸ C¸ELIK, ERHAN SET, AND AHMET OCAK AKDEMIR −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−r1 pZ1 0 ρϱp dρ +1 qZ1 0 |ℵ′(ρt + (1 −ρ)r)|qdρ +(s−t)ϱ+1 s−r1 pZ1 0 ρϱp dρ +1 qZ1 0 |ℵ′(ρt + (1 −ρ)s)|qdρ ≤(t−r)ϱ+1 s−r1 p(ϱp + 1) +1 qmax |ℵ′(t)|q,|ℵ′(r)|q +(s−t)ϱ+1 s−r1 p(ϱp + 1) +1 qmax |ℵ′(t)|q,|ℵ′(s)|q. So, the proof is completed. □ Theorem 2.4. Assume that Ψ:[r, s]→Ris a strictly increasing and positive function with a continuous derivative on [r, s]. Let ℵ: [r, s]→Rbe a differentiable function on (r, s),ℵ′∈L1[r, s], and r < s. If |ℵ′|is η-convex with respect to η, the following inequality holds: (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−r|ℵ′(r)| ϱ+ 1 +η(|ℵ′(t)|,|ℵ′(r)|) ϱ+ 2  +(s−t)ϱ+1 s−r|ℵ′(s)| ϱ+ 1 +η(|ℵ′(t)|,|ℵ′(s)|) ϱ+ 2 , where t∈[r, s],ϱ∈(0,1], and M(ϱ)>0. Proof. By using the identity that is given in Lemma 2.1, we obtain (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−rZ1 0 ρϱ|ℵ′(ρt + (1 −ρ)r)|dρ +(s−t)ϱ+1 s−rZ1 0 ρϱ|ℵ′(ρt + (1 −ρ)s)|dρ. Since |ℵ′|is η-convex, we get (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−rZ1 0 ρϱ[|ℵ′(r)|+ρη(|ℵ′(t)|,|ℵ′(r)|)] dρ SOME NOVEL INTEGRAL INEQUALITIES 7 +(s−t)ϱ+1 s−rZ1 0 ρϱ[|ℵ′(s)|+ρη(|ℵ′(t)|,|ℵ′(s)|)] dρ =(t−r)ϱ+1 s−r|ℵ′(r)| ϱ+ 1 +η(|ℵ′(t)|,|ℵ′(r)|) ϱ+ 2  +(s−t)ϱ+1 s−r|ℵ′(s)| ϱ+ 1 +η(|ℵ′(t)|,|ℵ′(s)|) ϱ+ 2  and the proof is completed. □ Theorem 2.5. Assume that Ψ:[r, s]→Ris a strictly increasing and positive function with a continuous derivative on [r, s]. Let ℵ: [r, s]→Rbe a differentiable function on (r, s),ℵ′∈L1[r, s], and r < s. If |ℵ′|qis η-convex with respect to η, where q≥1, the following inequality holds for modified AB-fractional integral operators: (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−r1 ϱp + 1 1 p|ℵ′(r)|q+η(|ℵ′(t)|q,|ℵ′(r)|q) 2 1 q +(s−t)ϱ+1 s−r1 ϱp + 1 1 p|ℵ′(s)|q+η(|ℵ′(t)|q,|ℵ′(s)|q) 2 1 q , where 1 p+1 q= 1,t∈[r, s],ϱ∈(0,1], and M(ϱ)>0. Proof. Using Lemma 2.1 and applying the H¨older inequality, we have (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−rZ1 0 ρϱp dρ 1 pZ1 0 |ℵ′(ρt + (1 −ρ)r)|qdρ 1 q +(s−t)ϱ+1 s−rZ1 0 ρϱp dρ 1 pZ1 0 |ℵ′(ρt + (1 −ρ)s)|qdρ 1 q . Since |ℵ′|qis η-convex, we obtain (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−rZ1 0 ρϱp dρ 1 pZ1 0 [|ℵ′(r)|q+ρη(|ℵ′(t)|q,|ℵ′(r)|q)] dρ 1 q 8 BARIS¸ C¸ELIK, ERHAN SET, AND AHMET OCAK AKDEMIR +(s−t)ϱ+1 s−rZ1 0 ρϱp dρ 1 pZ1 0 [|ℵ′(s)|q+ρη(|ℵ′(t)|q,|ℵ′(s)|q)] dρ 1 q =(t−r)ϱ+1 s−r1 ϱp + 1 1 p|ℵ′(r)|q+η(|ℵ′(t)|q,|ℵ′(r)|q) 2 1 q +(s−t)ϱ+1 s−r1 ϱp + 1 1 p|ℵ′(s)|q+η(|ℵ′(t)|q,|ℵ′(s)|q) 2 1 q . So, the proof is completed. □ Theorem 2.6. Assume that Ψ:[r, s]→Ris a strictly increasing and positive function with a continuous derivative on [r, s]. Let ℵ: [r, s]→Rbe a differentiable function on (r, s),ℵ′∈L1[r, s], and r < s. If |ℵ′|qis η-convex with respect to η, where q≥1, the following inequality holds: (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−r1 ϱ+ 11−1 q|ℵ′(r)|q ϱ+ 1 +η(|ℵ′(t)|q,|ℵ′(r)|q) ϱ+ 2  1 q +(s−t)ϱ+1 s−r1 ϱ+ 11−1 q|ℵ′(s)|q ϱ+ 1 +η(|ℵ′(t)|q,|ℵ′(s)|q) ϱ+ 2  1 q , where t∈[r, s],ϱ∈(0,1], and M(ϱ)>0. Proof. From Lemma 2.1, applying the power-mean inequality, we get (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−rZ1 0 ρϱdρ1−1 qZ1 0 ρϱ|ℵ′(ρt + (1 −ρ)r)|qdρ 1 q +(s−t)ϱ+1 s−rZ1 0 ρϱdρ1−1 qZ1 0 ρϱ|ℵ′(ρt + (1 −ρ)s)|qdρ 1 q . Since |ℵ′|qis η-convex, we have (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−rZ1 0 ρϱdρ1−1 qZ1 0 ρϱ[|ℵ′(r)|q+ρη(|ℵ′(t)|q,|ℵ′(r)|q)] dρ 1 q SOME NOVEL INTEGRAL INEQUALITIES 9 +(s−t)ϱ+1 s−rZ1 0 ρϱdρ1−1 qZ1 0 ρϱ[|ℵ′(s)|q+ρη(|ℵ′(t)|q,|ℵ′(s)|q)] dρ 1 q =(t−r)ϱ+1 s−r1 ϱ+ 11−1 q|ℵ′(r)|q ϱ+ 1 +η(|ℵ′(t)|q,|ℵ′(r)|q) ϱ+ 2  1 q +(s−t)ϱ+1 s−r1 ϱ+ 11−1 q|ℵ′(s)|q ϱ+ 1 +η(|ℵ′(t)|q,|ℵ′(s)|q) ϱ+ 2  1 q . So, the proof is completed. □ Theorem 2.7. Assume that Ψ:[r, s]→Ris a strictly increasing and positive function with a continuous derivative on [r, s]. Let ℵ: [r, s]→Rbe a differentiable function on (r, s),ℵ′∈L1[r, s], and r < s. If |ℵ′|qis η-convex with respect to η, where q≥1, the following inequality holds: (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−r1 p(ϱp + 1) +1 q|ℵ′(r)|q+η(|ℵ′(t)|q,|ℵ′(r)|q) 2 +(s−t)ϱ+1 s−r1 p(ϱp + 1) +1 q|ℵ′(s)|q+η(|ℵ′(t)|q,|ℵ′(s)|q) 2, where 1 p+1 q= 1,t∈[r, s],ϱ∈(0,1], and M(ϱ)>0. Proof. From Lemma 2.1, apply the Young inequality ab ≤ap p+bq q, we get (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)] −M(ϱ)Γ(ϱ) s−rhMAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(r)) + MAB ΨIϱ Ψ−1(t)(ℵ ◦ Ψ)(Ψ−1(s))i ≤(t−r)ϱ+1 s−r1 pZ1 0 ρϱp dρ +1 qZ1 0 |ℵ′(ρt + (1 −ρ)r)|qdρ +(s−t)ϱ+1 s−r1 pZ1 0 ρϱp dρ +1 qZ1 0 |ℵ′(ρt + (1 −ρ)s)|qdρ. Since |ℵ′|qis η-convex and by a simple computation, we have the desired result. □ Theorem 2.8. Assume that Ψ:[r, s]→Ris a strictly increasing and positive function with a continuous derivative on [r, s]. Let ℵ: [r, s]→Rbe a differentiable function on (r, s),ℵ′∈L1[r, s], and r < s. If |ℵ′|is η-concave with respect to η, the following inequality holds: (t−r)ϱ+ (s−t)ϱ s−rℵ(t) + 1−ϱ s−rΓ(ϱ)[ℵ(r) + ℵ(s)]