New generalized P\xf3lya\u2013Szeg\xf6 and \u010ceby\u0161ev type inequalities with general kernel and measure

S Iqbal,Thabet Abdeljawad,M Samraiz,G Rahman,M Adil Khan,Kottakkaran Sooppy Nisar

doi:10.1186/s13662-020-03134-6

Abstract

It is always attractive and motivating to acquire the generalizations of known results. In this article, we introduce a new class mathfrak{C(h)} of functions which can be represented in a form of integral transforms involving general kernel with σ-finite measure. We obtain some new Pólya–Szegö and Čebyšev type inequalities as generalizations to the previously proved ones for different fractional integrals including fractional integral of a function with respect to another function capturing Riemann–Liouville integrals, Hadamard fractional integrals, Katugampola fractional integral operators, and conformable fractional integrals. This new idea shall motivate the researchers to prove the results over a measure space with general kernels instead of special kernels.

Highlights

There are several problems in the mathematics and its related real world applications wherein fractional derivatives occupy an important place [1,2,3,4,5,6,7,8,9,10,11,12,13]
The fractional derivatives take a place in a way they can be described in different approaches, where these approaches can be used to explain a lot of essential real world problems
Analyzing the uniqueness of fractional ordinary and partial differential equations can be performed by employing fractional integral inequalities

Summary

Introduction

There are several problems in the mathematics and its related real world applications wherein fractional derivatives occupy an important place [1,2,3,4,5,6,7,8,9,10,11,12,13]. The fractional derivatives take a place in a way they can be described in different approaches, where these approaches can be used to explain a lot of essential real world problems. Many authors have utilized unique versions of such inequalities to study diverse classes of differential and integral equations. Pólya–Szegö introduced one of the most intensively studied inequalities in [43] stated as follows: χ12(ζ

The constant

Choosing β

If we choose β