Abstract
In this article, we have established the Hadamard inequalities for strongly convex functions using generalized Riemann–Liouville fractional integrals. The findings of this paper provide refinements of some fractional integral inequalities. Furthermore, the error bounds of these inequalities are given by using two generalized integral identities.
Highlights
IntroductionE above inequality is well-known as the Hadamard inequality. is inequality provides lower and upper estimates for integral average of a convex function
Let f: I ⟶ R be a convex function defined on an intervalI ⊂ R and x, y ∈ I, where x < y. en, the following inequality holds: x+y f ≤ y f(x) + f(y) f(v)dv ≤ 2 y− x x (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
E above inequality is well-known as the Hadamard inequality. is inequality provides lower and upper estimates for integral average of a convex function
Summary
E above inequality is well-known as the Hadamard inequality. is inequality provides lower and upper estimates for integral average of a convex function. E Hadamard inequality is studied for these integral operators of strongly convex functions, and Mathematical Problems in Engineering by using some integral identities, error bounds are established. E fractional versions of Hadamard inequality by Riemann–Liouville fractional integrals are given in the following theorems. If f is strongly convex function on [a, b] with modulus C ≥ 0, the following fractional integral inequalities hold:. E k-analogue of generalized Riemann–Liouville fractional integrals are defined as follows: Theorem 7 (see [35]). E findings of this paper are connected with results that are explicitly proved in [1, 2, 31, 35, 36, 40,41,42,43,44]
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