Some results on quantum Hahn integral inequalities

Abstract

In this paper the quantum Hahn difference operator and the quantum Hahn integral operator are defined via the quantum shift operator θnew fractional integral inequalities are established by using the quantum Hahn integral for one and two functions bounded by quantum integrable functions. The Hermite-Hadamard type of ordinary and fractional quantum Hahn integral inequalities as well as the Pólya-Szegö type fractional Hahn integral inequalities and the Grüss-Cebyšev type fractional Hahn integral inequality are also presented. MSC: Primary 39A10; 39A13; secondary 39A70 Keywords: Hahn difference operator; Hahn integral operator; Hahn difference inequalities; Hermite-Hadamard quantum Hahn integral inequality; Pólya-Szegö type fractional Hahn integral inequalities; Grüss-Cebyšev type fractional Hahn integral inequality 1 Introduction and preliminaries Let be f defined on an interval I ⊆ R containing ω 0 := ω 1-q . The Hahn difference operator D q,ω , introduced in [1], is defined as D q,ω f (t) = ⎧ ⎨ ⎩ f (qt+ω)-f (t) t(q-1)+ω , t = ω 0 , f (ω 0 ), t = ω 0 ,

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“…Because k ∈ (0, 1) by assumptions (22), therefore, Ψ is a contraction. Thus, it follows by Banach's contraction principle that the boundary value problem (4) has a unique solution.…”

“…In case λ(t) = λ, where λ is a constant, the condition (22) becomes λA < 1 and Theorem 3 takes the form of the following results.…”

Some trapezoid and midpoint type inequalities via fractional $(p,q)$-calculus

“…4 Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, 21588 Jeddah, Saudi Arabia. 5 International Center for Basic and Applied Sciences, 302029 Jaipur, India. 6 Department of Mathematics, Anand International College of Engineering, 302029 Jaipur, India.…”

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“…Asawasamrit et al [76] expounded the concept of qderivative over the interval ½l 1 , l 2 ⊂ ℝ and derived several inequalities on quantum analogues, for example, q-Cauchy-Schwarz inequality, q-Grüss-Čebyšev integral inequality, q-Grüss inequality, and other integral inequalities, by use of the convexity theory.…”