On the weighted fractional P\xf3lya\u2013Szeg\xf6 and Chebyshev-types integral inequalities concerning another function

Kottakkaran Sooppy Nisar,Sajid Iqbal,Dumitru Baleanu,Muhammad Samraiz,Gauhar Rahman

doi:10.1186/s13662-020-03075-0

Abstract

The primary objective of this present paper is to establish certain new weighted fractional Pólya–Szegö and Chebyshev type integral inequalities by employing the generalized weighted fractional integral involving another function Ψ in the kernel. The inequalities presented in this paper cover some new inequalities involving all other type weighted fractional integrals by applying certain conditions on omega (theta ) and Psi (theta ). Also, the Pólya–Szegö and Chebyshev type integral inequalities for all other type fractional integrals, such as the Katugampola fractional integrals, generalized Riemann–Liouville fractional integral, conformable fractional integral, and Hadamard fractional integral, are the special cases of our main results with certain choices of omega (theta ) and Psi (theta ). Additionally, examples of constructing bounded functions are also presented in the paper.

Highlights

The field of integral inequalities plays an essential role in the diverse domain
4 Chebyshev type weighted fractional integral inequalities we present Chebyshev type weighted fractional integral inequalities by using the Pólya–Szegö integral inequality given by Lemma 3.1 by employing weighted fractional integral (2.9)
7 Concluding remarks In this present investigation, we presented some new weighted fractional Pólya–Szegö and Chebyshev type integral inequalities by employing weighted fractional integral recently proposed by Jarad et al [14]

Summary

Introduction

The field of integral inequalities plays an essential role in the diverse domain. The mathematicians have investigated that it is mainly a powerful tool for the improvement of both applied and pure mathematics. In [8], the authors established Grüss type integral inequalities by employing the classical fractional integrals. Certain new integral inequalities for the Riemann–Liouville (R-L) fractional integrals can be found in the work of Dahmani [6]. Nisar et al [27] performed Gronwall inequalities with applications. Rahman et al [42] gave certain inequalities for (k, ρ)-fractional integrals. Ostrowski type inequalities connecting local fractional integrals were found in [50]. Sarikaya et al [51] developed generalized (k, s)-fractional integrals with applications. In [52], Set et al introduced Grüss type inequalities by employing generalized k-fractional integrals. Nisar et al [29] gave some new generalized fractional integral inequalities

Objectives

Conclusion