Analysis of ( P , m ) $(P,\mathrm{m})$ -superquadratic function and related fractional integral inequalities with applications

Abstract

Abstract In the present work we establish for the first time a class of ( P , m ) $(P,\mathrm{m})$ -superquadratic functions and look into its features. Using them, we come up with the Jensen and Hermite–Hadamard inequalities, as well as the fractional versions of Hermite–Hadamard inequalities with respect to Riemann–Liouville fractional integral operators. The findings are confirmed by certain numerical calculations and graphical depictions that take a few appropriate examples into account. The study is enhanced by the addition of applications of special means, moments of random variables, and modified Bessel functions of the first kind. This is achieved by considering new functions pertaining to the uniform probability density function and taking the modified Bessel functions of the first kind into consideration. The new results clearly provide extensions and improvements of the work given in the literature.

Keywords

• ( P , m ) $(P,\mathrm{m})$ -superquadratic function
• Jensen inequality
• Hermite–Hadamard inequality
• Riemann–Liouville fractional integral operators
• Moment of random variable
• Modified Bessel function of the first kind