Fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex functions

1 School of Mechanical and Electrical, Hubei Polytechnic University, Huangshi, 435003, China 2 Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan 3 Department of Mathematics, GC University, Lahore, Pakistan 4 Department of Mathematics, University of Sargodha, Sargodha, Pakistan 5 Department of Mathematics, Huzhou University, Huzhou, 313000, P. R. China 6 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha, 410114, P. R. China 7 School of Mechanical and Electrical, Hubei Polytechnic University, Huangshi, 435003, China


Introduction and preliminary results
The theory of inequalities of convex functions is part of the general subject of convexity since a convex function is one whose epigraph is a convex set. Nonetheless it is a theory important per se, which touches almost all branches of mathematics. Probably, the first topic who make necessary the encounter with this theory is the graphical analysis. With this occasion we learn on the second derivative test of convexity, a powerful tool in recognizing convexity. Then comes the problem of finding the extremal values of functions of several variables and the use of Hessian as a higher dimensional generalization of the second derivative. Passing to optimization problems in infinite dimensional spaces is the next step, but despite the technical sophistication in handling such problems, the basic ideas are pretty similar with those underlying the one variable case.
The objective of this paper is to present the fractional Hadamard and Fejér-Hadamard inequalities in generalized forms. We start from the integral operators containing generalized Mittag-Leffler function defined by Prabhaker in [25]. Definition 1.1. Let σ, τ, ρ be positive real numbers and ω ∈ R. Then the generalized fractional integral operators containing Mittag-Leffler function ρ σ,τ,ω,a + f and ρ σ,τ,ω,b − f for a real valued continuous function f are defined by: where the function E ρ σ,τ (t) is the generalized Mittag-Leffler function; E ρ σ,τ (t) = ∞ Fractional integral operators associated with generalized Mittag-Leffler function play a vital role in fractional calculus. Different fractional integral operators have different types of properties and these integral operators may be singular or non-singular depending upon their kernels. For example, the global Riemann Liouville integral is a singular integral operator but the singularity is integrable. Some new models [2,7] have been designed due to the non-singularity of their defining integrals. Fractional integral operators are useful in the generalization of classical mathematical concepts. Fractional integral operators are very fruitful in obtaining fascinating and glorious results, for example fractional order systems and fractional differential equations are used in physical and mathematical phenomena. Many inequalities like Hadamard are studied in the context of fractional calculus operators, see [1,6,8,20,28].
Further fractional integral operators containing an extended generalized Mittag-Leffler function in their kernels are defined as follows: Then the generalized fractional integral operators ρ,r,k,c σ,τ,δ,ω,a + f and ρ,r,k,c σ,τ,δ,ω,b − f are defined by: is the extended generalized Mittag-Leffler function.
be the functions such that f be positive and f ∈ L 1 [a, b] and g be a differentiable and strictly increasing. Also let ω, τ, δ, ρ, c ∈ C, (τ), (δ) > 0, (c) > (ρ) > 0 with p ≥ 0, σ, r > 0 and 0 < k ≤ r + σ. Then for x ∈ [a, b] the integral operators are defined by: ). (1.11) The following remark provides some connection of Definition 1.5 with already known operators: Remark 1. (i) If we take p = 0 and g(x) = x in equation (1.10), then it reduces to the fractional integral operators defined by Salim and Faraj in [27].
(iii) If we set p = 0, δ = r = 1 and g(x) = x in (1.10), then it reduces to integral operators containing extended generalized Mittag-Leffler function introduced by Srivastava and Tomovski in [29].
More than a hundred years ago, the mathematicians introduced the convexity and they established a lot of inequalities for the class of convex functions. The convex functions are playing a significant and a tremendous role in fractional calculus. Convexity has been widely employed in many branches of mathematics, for instance, in mathematical analysis, optimization theory, function theory, functional analysis and so on. Recently, many authors and researchers have given their attention to the generalizations, extensions, refinements of convex functions in multi-directions.
The m-convex function is a close generalization of convex function and its concept was introduced by Toader [30].
If we take m = 1, we get the definition for convex function. An m-convex function need not be a convex function. 16 17 -convex but it is not m-convex for m ∈ ( 16 17 , 1]. A lot of results and inequalities pertaining to convex, m-convex and related functions have been produced (see, [11][12][13][14][15][16][17][18][19][20] and references therein). Many fractional integral inequalities like Hadamard and Fejér-Hadamard are very important and researchers have produced their generalizations and refinements (see, [5] and references therein). Fractional inequalities have many applications, for instance, the most fruitful ones are used in establishing uniqueness of solutions of fractional boundary value problems and fractional partial differential equations. For instance the following Hadamard inequality is given in [21]: , then the following inequalities for the extended generalized fractional integrals hold: ; p) In the upcoming section we will derive the Hadamard inequality for m-convex functions by means of fractional integrals (1.10) and (1.11). This version of the Hadamard inequality gives at once the Hadamard inequalities quoted in Section 2. Further we will establish the Fejér-Hadamard inequality for these operators of m-convex functions which will provide the corresponding inequalities proved in [31]. Moreover in Section 3 by establishing two identities error estimations of the Hadamard and the Fejér-Hadamard inequalities are obtained.

Main results
, be the functions such that f be positive and f ∈ L 1 [a, b], g be differentiable and strictly increasing. If f be m-convex m ∈ (0, 1] and g(a) < mg(b), then the following inequalities for fractional operators (1.10) and (1.11) hold: f g(a) + mg(b) 2 g Υ ρ,r,k,c σ,τ,δ,ω ,a + 1 (g −1 (mg(b)); p) Proof. By definition of m-convex function f , we have Further from (2.1), one can obtain the following integral inequality: 2), we get the following inequality: Also by using the m-convexity of f , one can has This leads to the following integral inequality: Again by setting g(x) = tg(a) + m(1 − t)g(b), g(y) = tg(b) + (1 − t) g(a) m in (2.5) and after calculation, we get ) + m f g(a) m 2 g Υ ρ,r,k,c σ,τ,δ,m σ ω ,b − 1 g −1 g(a) m ; p . The following theorem gives the Fejér-Hadamard inequality for m-convex functions.
• In Theorem 2.2, if we put m = 1, then we get [31,Theorem 3.2]. • In Theorem 2.2, if we put g = I and p = 0, then we get results of [3].
• In Theorem 2.2, if we put g = I, then we get results of [4].
By using Lemma 3.2, we prove the following theorem.