Trapezium-Type Inequalities for Raina’s Fractional Integrals Operator Using Generalized Convex Functions

• Main Authors: Miguel Vivas-Cortez, Artion Kashuri, Jorge E. Hernández Hernández
• Format: Article
• Language: English
• Published: MDPI AG 2020-06-01
• Series: Symmetry
• Subjects: Hermite–Hadamard inequality; Raina’s fractional integral operator; Hölder inequality; power mean inequality; generalized convexity
• Online Access: https://www.mdpi.com/2073-8994/12/6/1034

The authors have reviewed a wide production of scientific articles dealing with the evolution of the concept of convexity and its various applications, and based on this they have detected the relationship that can be established between trapezoidal inequalities, generalized convex functions, and special functions, in particular with the so-called Raina function, which generalizes other better known ones such as the hypergeometric function and the Mittag–Leffler function. The authors approach this situation by studying the Hermite–Hadamard inequality, establishing a useful identity using Raina’s fractional integral operator in the setting of <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula>-convex functions, obtaining some integral inequalities connected with the right-hand side of Hermite–Hadamard-type inequalities for Raina’s fractional integrals. Various special cases have been identified.