Refinements and Generalizations of Some Fractional Integral Inequalities via Strongly Convex Functions

Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, Pakistan Department of Business Administration, Gyeongsang National University, Jinju 52828, Republic of Korea Department of Refrigeration and Air Conditioning Engineering, Chonnam National University, Yeosu 59626, Republic of Korea School of Mathematics and Statistics, Northeast Normal University, Changchun 130000, China


Introduction
Let f: I ⟶ R be a convex function defined on an interval I ⊂ R and x, y ∈ I, where x < y.
en, the following inequality holds: (1) e above inequality is well-known as the Hadamard inequality. is inequality provides lower and upper estimates for integral average of a convex function. Since the appearance of this result in literature, it has drawn attention of many mathematicians of recent age and it is one of the most extensively studied results for convex functions. In [1,2], Sarikaya et al. have studied it via Riemann-Liouville fractional integrals of convex functions. After these versions of Hadamard inequality, many researchers were motivated and elegantly produced fractional inequalities using different types of fractional integrals. Also, many new classes of functions have been introduced in the establishment of fractional Hadamard inequalities; for details, we refer the readers to [3][4][5][6][7][8][9][10][11].
Fractional calculus studies the integrals and derivatives of any arbitrary order, real or complex. Its history begins at the end of seventeenth century, when G. W. Leibniz and Marquis de l'Hospital in 1695 introduced it for first time by discussing the differentiation of functions of order 1/2. However, it experienced a rapid growth over the short span of time. For example, Lagrange, Laplace, Lacroix, Fourier, Abel, Liouville, Riemann, Green, Holmgren, Grunwald, Letnikov, Sonin, Laurent, Nekrassov, Krug, and Weyl made their major contributions to establish a solid foundation of fractional calculus (see [12][13][14] and references there in). Fractional integral and derivative operators are the key factors in the development of fractional calculus. Recently, the generalizations [15][16][17], extensions [18][19][20], and applications [21][22][23] for fractional operators have been made by many researchers in mathematics, fluid mechanics [24][25][26], biological population models [27], and numerical methods [28].
Our aim in this paper is to utilize generalized Riemann-Liouville fractional integrals with monotonically increasing function. e Hadamard inequality is studied for these integral operators of strongly convex functions, and also, by using some integral identities, error bounds are established. Next, we give the definition of strongly convex function introduced by Polyak [29] (see also [30]). Definition 1. Let D be a convex subset of X, (X, ‖.‖) be a normed space. A function f: D ⊂ X ⟶ R will be called strongly convex function with modulus C ≥ 0 if holds ∀x, y ∈ D ⊆ X, t ∈ [0, 1]. For C � 0, (2) gives the definition of convex function.
In the following, we give the definition of Riemann-Liouville fractional integrals.
en, left-sided and rightsided Riemann-Liouville fractional integrals of a function f of order μ where R(μ) > 0 are defined as follows: e fractional versions of Hadamard inequality by Riemann-Liouville fractional integrals are given in the following theorems.
Theorem 1 (see [1]). Let f: [a, b] ⟶ R be a positive function with 0 ≤ a < b and f ∈ L 1 [a, b]. If f is a convex function on [a, b], then the following fractional integral inequalities hold: with α > 0.
Theorem 7 (see [35]). Let f: , then the following inequalities for k-fractional integrals hold: Theorem 8 (see [36]). Let f: [a, b] ⟶ R be a positive function with 0 ≤ a < b. If f is a convex function on [a, b], then the following inequalities for k-fractional integrals hold: Theorem 9 (see [35]). Let f: , then the following inequality for k-fractional integrals hold: In the following, we give the definition of generalized Riemann-Liouville fractional integrals by a monotonically increasing function: Definition 4 (see [37]). Let f: [a, b] ⟶ R be an integrable function. Also, let ψ be an increasing and positive function on (a, b], having a continuous derivative ψ ′ on (a, b). e left-sided and right-sided fractional integrals of a function f with respect to another function ψ on [a, b] of order μ where R(μ) > 0 are defined by If ψ is identity function, then (17) and (18) coincide with (3) and (4).
e k-analogue of generalized Riemann-Liouville fractional integrals are defined as follows: Definition 5 (see [38]). Let f: [a, b] ⟶ R be an integrable function. Also, let ψ be an increasing and positive function on
In Section 2, we establish Hadamard inequalities for generalized Riemann-Liouville fractional integrals of strongly convex functions. e particular cases are given as consequences of these inequalities which are connected with already published results. In Section 3, by using two integral identities for generalized fractional integrals, the error bounds of fractional Hadamard inequalities are established. e findings of this paper are connected with results that are explicitly proved in [1,2,31,35,36,[40][41][42][43][44]].

Main Results
Also, suppose that f is strongly convex function on [a, b] with modulus C ≥ 0, ψ is an increasing and positive monotone function on (a, b], having a continuous derivative ψ ′ (x) on (a, b). en, for k > 0, the following k-fractional integral inequalities hold: with α > 0. Proof.

Corollary 1.
Under the assumption of eorem 10 with k � 1 in (21), the following inequality holds: Corollary 2. Under the assumption of eorem 10 with ψ as identity function in (21), the following inequality holds:  (a, b). en, for k > 0, the following k-fractional integral inequalities hold: with α > 0.

Corollary 3.
Under the assumption of eorem 11 with C � 0 in (33), the following inequality holds:

Corollary 4.
Under the assumption of eorem 11 with k � 1 in (33), the following inequality holds: Corollary 5. Under the assumption of eorem 11 with ψ as identity function in (33), the following inequality holds:

Error Bounds of Hadamard Inequalities for Strongly Convex Functions
In this section, we provide the error bounds of fractional Hadamard inequalities using generalized Riemann-Liouville fractional integrals via strongly convex functions. Estimations here are further refined as compared to those already established for convex functions. e following lemma is useful to prove the next result.
Proof. From Lemma 1 and strongly convexity of |f ′ |, we have It can be noted that erefore, (47) implies From which after a little computation, one can get (46). □ Remark 3. Under the assumption of eorem 12, one can get the following results: (i) If k � 1 and ψ is identity function in (46), then eorem 6 is obtained. (ii) If C � 0 and ψ is identity function in (46), then eorem 9 is obtained.

Corollary 6.
Under the assumption of eorem 12 with C � 0 in (46), the following inequality holds: Corollary 7. Under the assumption of eorem 12 with k � 1 in (46), the following inequality holds:

Mathematical Problems in Engineering
Corollary 8. Under the assumption of eorem 12 with ψ as identity function in (46), the following inequality holds: We now derive a new fractional integral identity for fractional integrals (19) and (20).  (a, b). en, for k > 0, the following identity holds: with α > 0.

Corollary 9.
Under the assumption of Lemma 2 with k � 1 in (54), the following identity holds: (60) Using above lemma, we give the following error bounds of the k-fractional Hadamard inequality.
Theorem 13. Let f: I ⟶ R be a differentiable mapping on (a, b) with a < b. Also, suppose that |f ′ | q is strongly convex function on [a, b] with modulus C ≥ 0 for q ≥ 1, and ψ is an increasing and positive monotone function on (a, b], having a continuous derivative ψ ′ (x) on (a, b). en, for k > 0, the following k-fractional integral inequalities hold: Mathematical Problems in Engineering with α > 0.
Proof. From Lemma 2 and strongly convexity of |f ′ |, let q � 1, we have Now, for q > 1, we proceed as follows. From Lemma 2 and using power mean inequality, we get Strongly convexity of |f ′ | q gives which after a little computation gives the required result. □ Remark 5. Under the assumption of eorem 13, one can get the following results: (i) If C � 0 and ψ is identity function in (61), then the inequality ( eorem 3.1) stated in [36] is obtained.