Chebyshev inequality on conformable derivative

Aysun SELÇUK KIZILSU,Ayşe Feza GÜVENİLİR

doi:10.31801/cfsuasmas.773392

Abstract

Integral inequalities are very important in applied sciences. Chebyshev's integral inequality is widely used in applied mathematics. First of all, some necessary definitions and results regarding conformable derivative are given in this article. Then we give Chebyshev inequality for simultaneously positive (or negative) functions using the conformable fractional derivative. We used the Gronwall inequality to prove our results, unlike other studies in the literature.

Highlights

Various de...nitions are given in the literature for fractional derivatives [8, 14, 17, 20]
In [12], a new fractional derivative that is known as conformable derivative has been de...ned by Khalil
This new fractional derivative based on classical limit definition

Summary

Introduction

Various de...nitions are given in the literature for fractional derivatives [8, 14, 17, 20]. In [12], a new fractional derivative that is known as conformable derivative has been de...ned by Khalil. The product rule, the division rule, Rolle theorem and mean value theorem for this new de...nition of fractional derivative. They de...ned the fractional integral of order 0 < 1 only. In [1], de...nition of left and right conformable fractional integrals of any order > 0 has been given by Abdeljawad. He gave chain rule, linear di¤erential systems, Laplace transforms and exponential functions on a fractional version. For monotonicity and convexity results for fractional integrals and some of their application we recommend the readers to refer the literature [18, 13, 7, 5, 19]

Preliminaries

Main Result