Fractional Hadamard and Fejér-Hadamard Inequalities Associated with Exponentially <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:mfenced></mml:math>-Convex Functions
Shuya Guo, Yu‐Ming Chu, Ghulam Farid, Sajid Mehmood, Waqas Nazeer
Abstract
The aim of this paper is to present the fractional Hadamard and Fejér-Hadamard inequalities for exponentially <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:mfenced></mml:math>-convex functions. To establish these inequalities, we will utilize generalized fractional integral operators containing the Mittag-Leffler function in their kernels via a monotone function. The presented results in particular contain a number of fractional Hadamard and Fejér-Hadamard inequalities for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>s</mml:mi></mml:math>-convex, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>m</mml:mi></mml:math>-convex, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:mfenced></mml:math>-convex, exponentially convex, exponentially <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>s</mml:mi></mml:math>-convex, and convex functions.