Chebyshev type inequalities with fractional delta and nabla h-sum operators

Serkan Asliyüce,Ayşe Feza Güveni̇li̇r

doi:10.31801/cfsuasmas.455707

Abstract

The aim of this study is to establish new discrete inequalities for synchronous functions using fractional order delta and nabla h-sum operators. We give examples to illustrate our results.

Highlights

Z1 f (x)g(x)dx f (x)dx g(x)dx: Since generalizations and extensions of such type inequality have appeared in the literature, see [13, 14, 17, 18, 24] and references cited therein
To see that, (i) Taking h = 1 in Theorems 12, 13 and 16, we obtain the inequalities given by Bohner and Ferreira [11], (ii) Taking h = 1 in Theorems 18, 20 and 22, we get the inequalities introduced in [3]
Declaration of Competing Interests The authors declare that they have no competing interests

Summary

Introduction

In 1882, P.L. Chebyshev [12] proved the following inequality: Let f and g be two integrable functions on [0; 1]: If both functions are simultaneously increasing or decreasing for the same values of x 2 [0; 1]; . If one function is increasing and the other decreasing for the same values of x 2 [0; 1]; . Let f and g be two synchronous functions on [0; 1) : for all t > 0; > 0; we have where Ja is as. The fractional order discrete Chebyshev type inequalities are studied in [3, 11]. To establish the fractional analogues of Chebyshev inequality, in discrete case, we will use the delta and nabla h-sum operators de...ned in [9,16,20,23]

Preliminaries and basic results

Delta Chebyshev’s inequality

Conclusions