Refinements of Some Integral Inequalities forφ-ConvexFunctions

Convex functions play an important role in the formation of new definitions of related functions which help to give the generalization of classical results. -erefore, in recent years, many generalizations of convex functions are defined and utilized to study the Hadamard and other well-known inequalities (see [1–9]). In this paper, we deal with the strongly φ-convex functions to study the bounds of unified integral operators. -e obtained results are compared with already known results. First, we give some definitions of functions which are necessary for the findings of this paper.


Introduction and Preliminaries
Convex functions play an important role in the formation of new definitions of related functions which help to give the generalization of classical results. erefore, in recent years, many generalizations of convex functions are defined and utilized to study the Hadamard and other well-known inequalities (see [1][2][3][4][5][6][7][8][9]). In this paper, we deal with the strongly φ-convex functions to study the bounds of unified integral operators. e obtained results are compared with already known results.
First, we give some definitions of functions which are necessary for the findings of this paper.
Definition 1 (see [7]). A function f: I ⟶ R is said to be convex on I if holds for all u, v ∈ I and ς ∈ [0, 1], where I⊆R is an interval. Reverse of inequality (1) defines f as concave function.
Definition 2 (see [10]). A function f: I ⟶ R is said to be strongly convex with modulus λ > 0 if holds for all u, v ∈ I and ς ∈ [0, 1].
Definition 3 (see [3]). A function f: I ⟶ R is said to be φ-convex on I if holds for all u, v ∈ I and ς ∈ [0, 1], where φ is a bifunction.
Definition 4 (see [2]). A function f: I ⟶ R is said to be strongly φ-convex on I if holds for all u, v ∈ I and ς ∈ [0, 1], λ ≥ 0, where φ is a bifunction. It is to be noted that for φ(x, y) � x − y, strongly φ-convex function reduces to strongly convex function. Farid in [11] defined the unified integral operators (5) and (6) and has proved the continuity and the boundedness of these integral operators. e aim of this paper is the study of integral inequalities for strongly φ-convex functions via be a positive φ-convex function and g: [u, v] ⟶ R be differentiable and strictly increasing function. Also, let Ψ/x be an increasing function on [u, v], η, α, ξ, c, ζ ∈ C, p, μ, ], δ ≥ 0, 0 < k ≤ δ + μ, and and φ(x, y) � x + y, then the following result holds: Also, the following result holds for the convolution of functions f and g.

Main Results
roughout this section, we have adopted the following notations: Theorem 4. If f is positive strongly φ-convex function with modulus λ ≥ 0, along with other assumptions of eorem 1, then we have where I d is the identity function. Proof. For the kernel defined in (7) and the strongly φ-convexity of the function f on [u, x], the following inequalities hold, respectively: e aforementioned inequalities are used to obtain the following integral inequality:

Journal of Mathematics 3
In view of Definition 5 and applying integration by parts, from inequality (15), we get the following upper bound of the right-sided unified integral operator: Again for the kernel defined in (7) and the strongly φ-convexity of the function f on (x, v], the following inequalities hold, respectively: e aforementioned inequalities (17) and (18) are used to obtain the following integral inequality: In view of Definition 5 and applying integration by parts, from inequality (19), we get the following upper bound of the left-sided unified integral operator: Inequality (12) will be obtained by combining (16) and (20). □ Corollary 1. By setting μ � ] in (12), we get , then we will get the refinement of (8). (21), we get the result for strongly convex function. For φ(x, y) � x − y and λ � 0 in (21), we get the result of eorem 8 in [5].
We will use the following lemma for our next result.
and φ(x, y) � x + y in addition with the assumptions of eorem 4. en, the following inequality holds:

Journal of Mathematics
Proof. For the kernel defined in equation (7) and the strongly φ-convexity of the function f on [u, v], the following inequalities hold, respectively: e aforementioned inequalities are used to obtain the following integral inequality: In view of Definition 5, applying integration by parts, and using φ(x, y) � x + y, from inequality (27), we get the following upper bound of the left-sided unified integral operator: Also, the following inequality holds: e aforementioned inequalities (26) and (29) are used to obtain the following integral inequality: 6

Journal of Mathematics
In view of Definition 5 and applying integration by parts, from inequality (30), we get the following upper bound of the right-sided unified integral operator: Now, using Lemma 1, we can write In view of Definition 5 and φ(x, y) � x + y, from (32), we get the following upper bound of the left-sided unified integral operator: Also, from Lemma 1, we can write In view of Definition 5 and φ(x, y) � x + y, from (34), we get the following upper bound of the right-sided unified integral operator: Journal of Mathematics Inequality (24) will be obtained by using (28), (31), (33), and (35). □ Remark 3. For λ � 0 in (24), we get (9) of eorem 2; if 2I(u, v; I d g) > (u + v)I(u, v; g), then we will get refinement of (9). For φ(x, y) � x − y in (24), we get the result for strongly convex function.
For φ(x, y) � x − y and λ � 0 in (24), we get the result of eorem 22 in [5]. Theorem 6. If |f ′ | is strongly φ-convex with modulus λ ≥ 0 along with other assumptions of eorem 3, then the inequality and I d is the identity function. Proof. Using strongly φ-convexity of |f ′ | over [u, x] gives Using absolute value property, we can write 8 Journal of Mathematics e aforementioned inequality (13) and second inequality of (40) are used to obtain the following integral inequality: In view of (37) and applying integration by parts, from inequality (41), we get the following upper bound: Also, inequality (13) and the first inequality of (40) are used to obtain the following integral inequality: Inequalities (17), (38), and (44) are used to obtain the following upper bounds: Inequality (36) will be obtained by using (42)-(46). □ Corollary 2. By setting μ � ] in (36), we get the following inequality: Remark 4. For λ � 0 in (36), we get inequality (10) of eorem 3; if 2I(u, x; I d g) > (x + u) I(u, x; g) and 2I(x, v; I d g) > (v + x)I(x, v; g), then we will get the refinement of (10). For φ(x, y) � x − y in (47), we get the result for strongly convex function. For φ(x, y) � x − y and λ � 0 in (47), we get the result of eorem 25 in [5].

Results for Fractional Integral Operators
In this section, we give the bounds of some of the fractional integral operators which will be deduced from the results of Section 2.