Abstract
The aim of this paper is to study unified integral operators for generalized convex functions namely \((\alpha,h-m)\)-convex functions. We obtained upper as well as lower bounds of these integral operators in diverse forms. The results simultaneously hold for many kinds of well known fractional integral operators and for various kinds of convex functions.
Highlights
The notion named convexity has applications in almost all branches of mathematics for instance in mathematical analysis, optimization theory, mathematical statistics, graph theory etc
The aim of this paper is to study unified integral operators for generalized convex functions namely (α, h − m)-convex functions
The results simultaneously hold for many kinds of well known fractional integral operators and for various kinds of convex functions
Summary
The notion named convexity has applications in almost all branches of mathematics for instance in mathematical analysis, optimization theory, mathematical statistics, graph theory etc. A fractional integral operator containing an extended generalized Mittag-Leffler function in its kernel is defined as follows: Definition 3. For suitable settings of functions φ, g and certain values of parameters included in Mittag-Leffler function, several recently defined known fractional integrals studied in [9–11,14,16–25] can be reproduced, see [26, Remarks 6 and 7]. One can note that the definitions of (h − m)-convex and (α, m)-convex functions can be obtained by setting α = 1 and h(t) = t respectively in (14) Motivated by this generalized convexity we are interested to investigate the bounds of the sum of left and right sided integral operators for these functions. By using condition of symmetry, two sided Hadamard type bounds are obtained and by using (α, h − m)-convexity of function | f | and by defining an integral operator for convolution of two functions further bounds are studied. The reverse of inequality holds when g and φ I are of opposite monotonicity
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