Refinements of Some Integral Inequalities for (s, m)-Convex Functions

Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology,, Changsha 410114, China Faculty of Civil Engineering, Architecture and Geodesy, University of Split, Matice Hrvatske 15, 21000 Split, Croatia Department of Business Administration, Gyeongsang National University, Jinju 52828, Republic of Korea RUDN University, Moscow, Russia Center for General Education, China Medical University, Taichung 40402, Taiwan


Introduction
e following fractional integral operator is the well-known Riemann-Liouville fractional integral operator.
Next, generalizations of Riemann-Liouville fractional integral operators are given.
Definition 2 (see [2]). Let f: [a, b] ⟶ R be an integrable function. Also, let g be an increasing and positive function on (a, b], having a continuous derivative g ′ on (a, b). e left-sided and the right-sided fractional integrals of a function f with respect to another function g on [a, b] of order μ where R(μ) > 0 are defined by where Γ(·) is the gamma function.
A k-analogue of the above definition is given as follows.
Definition 3 (see [3]). Let f: [a, b] ⟶ R be an integrable function. Also, let g be an increasing and positive function on (a, b], having a continuous derivative g ′ on (a, b). e left-sided and right-sided fractional integrals of a function f with respect to another function g on [a, b] of order μ where R(μ), k > 0 are defined by where Γ k (x) � ∞ 0 t x− 1 e − (t k /k) dt, R(x) > 0. e following integral operator is given in [4].
Definition 4. Let f, g: [a, b] ⟶ R, 0 < a < b, be the functions such that f be positive and f ∈ L 1 [a, b] and g be differentiable and strictly increasing. Also, let ϕ/x be an increasing function on [a, ∞). en, for x ∈ [a, b], the left and right integral operators are defined by where K g (x, y; ϕ) � ((ϕ(g(x) − g(y)))/(g(x) − g(y))).
A fractional integral operator containing an extended generalized Mittag-Leffler function in its kernel is defined as follows.
Definition 7 (see [24]). A function f: I ⟶ R is said to be a convex function if the inequality holds for all a, b ∈ I and t ∈ [0, 1]. e concept of a strongly convex function is defined as follows.
Definition 8 (see [25]). Let I be a nonempty convex subset of a normed space. A real-valued function f is said to be strongly convex with modulus λ ≥ 0 on I if for each a, b ∈ I and t ∈ [0, 1], we have

Mathematical Problems in Engineering
A generalization of the convex function defined on the right half of the real line is called the s-convex function, and it is given as follows.
Definition 9 (see [26]). Let s ∈ [0, 1]. A function f: [0, ∞) ⟶ R is said to be an s-convex function in the second sense if holds for all a, b ∈ [0, ∞) and t ∈ [0, 1]. e notion of the m-convex function and strongly m-convex function is defined as follows.
Definition 12 (see [29]). A function f: e notion of the strongly (s, m)-convex function is defined as follows.
Using strongly (s, m)-convexity and utilizing fractional operators (6) and (7), some fractional integral inequalities are obtained as in [31]. e following result provides the bound of sum of left and right fractional integrals (6) and (7) for strongly (s, m)-convex functions at an arbitrary point.
Theorem 1 (see [31]). Let f: [a, b] ⟶ R be a real-valued function. If f is positive and strongly (s, m)-convex, then for α, β ≥ 1, the following fractional integral inequality holds: e following Hadamard-type inequality holds for generalized fractional integral operators for strongly (s, m)-convex functions.

Mathematical Problems in Engineering
In the following, using the strongly (s, m)-convexity of |f ′ |, a modulus inequality is obtained.
Theorem 3 (see [31]). Let f: [a, b] ⟶ R be a real-valued function. If f is differentiable and |f ′ | is strongly (s, m)-convex, then for α, β ≥ 1, the following fractional integral inequality holds: In [32], we studied the properties of the kernel given in (13). Here, we are interested in the following property. P: let g and ϕ/x be increasing functions. en, for μ,α,l , g; ϕ) satisfies the following inequality: is can be obtained from the following two straightforward inequalities: e reverse of inequality (13) holds when g and ϕ/x are decreasing. e upcoming section contains the results for unified integral operators dealing with the bounds of several fractional integral operators in a compact form by utilizing strongly (s, m)-convex functions. A compact version of the Hadamard inequality is presented, and also a modulus inequality is given for the differentiable function f such that |f ′ | is a strongly (s, m)-convex function. In the whole paper, we will use

Main Results
e following result provides the upper bound of unified integral operators.  (11) and (12), the following inequality holds:

Mathematical Problems in Engineering
Proof. By (P), the following inequalities hold: For a strongly (s, m)-convex function, the following inequalities hold for a < t < x and x < t < b, respectively: From (28) and (30), one can have i.e., On the other hand, from (29) and (31) Mathematical Problems in Engineering 5 i.e., By adding (33) and (35), (27) can be obtained.

Lemma 1. Let f: [a, mb] ⟶ R be a strongly (s, m)-convex
, m ≠ 0, then the following inequality holds: In the literature, many mathematicians have established many types of Hadamard inequalities, and for their generalizations, see [39][40][41][42]. is also motivates us to introduce the more generalized forms of Hadamard-type inequalities. So, by the help of the abovementioned lemma, the following result provides generalized Hadamard inequality for strongly (s, m)-convex functions.

Theorem 5.
Under the assumptions of eorem 4, in addition to f(x) � f((a + mb − x)/m), the following inequality holds: Proof. By (P), the following inequalities hold: A strongly (s, m)-convex function satisfying the following inequalities hold for a < x < b: From (39) Further, the aforementioned inequality takes the form which involves Riemann-Liouville fractional integrals in the right-hand side, and thus we have upper bound of the unified left-sided integral operator (2) as follows: 2I a, b, I d g − (a + b)I(a, b, g) .