Generalized k-Fractional Integral Operators Associated with Pólya-Szegö and Chebyshev Types Inequalities

Tao Yan,Ghulam Farid,Kamsing Nonlaopon,Sajid Mehmood,Zhiqiang Zhang

doi:10.3390/fractalfract6020090

Tao Yan, Ghulam Farid + Show 3 more

Open Access

https://doi.org/10.3390/fractalfract6020090

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Abstract

Inequalities related to derivatives and integrals are generalized and extended via fractional order integral and derivative operators. The present paper aims to define an operator containing Mittag-Leffler function in its kernel that leads to deduce many already existing well-known operators. By using this generalized operator, some well-known inequalities are studied. The results of this paper reproduce Chebyshev and Pólya-Szegö type inequalities for Riemann-Liouville and many other fractional integral operators.

Highlights

Integral and derivative operators of fractional order are simple and important tools to generalize the classical theories and well-known problems related to integer order derivatives and integrals
Many modern subjects in different fields of mathematics, engineering, and sciences have been introduced due to the applications of fractional derivatives and integrals. These days, fractional integral/derivative operators are very frequently considered by the researchers working on mathematical inequalities to extend the classical literature
Fractional integral operators are very useful in the advancement of mathematical inequalities

Summary

Introduction

Integral and derivative operators of fractional order are simple and important tools to generalize the classical theories and well-known problems related to integer order derivatives and integrals. The well-known Pólya-Szegö inequality gives the estimation of quotient in terms of the Chebyshev inequality for bounded functions These inequalities have been studied for Riemann-Liouville and other fractional integral operators in [10,14,15,16,17,18,19,20]. Where the positive and integrable functions f and g satisfy u ≤ f (τ ) ≤ U, v ≤ g(τ ) ≤ V, for all τ ∈ [ a, b] and constants u, U, v, V ∈ R Another famous inequality which will be useful to obtain our main results is the Pólya-Szegö inequality [4], stated as follows:. The presented results are the generalizations of the results which are already published in [10,14,19]

Fractional Integral Operators

Pólya-Szegö and Chebyshev Type Inequalities for Generalized k-Fractional

Conclusions