Some New Simpson’s and Newton’s Formulas Type Inequalities for Convex Functions in Quantum Calculus

Abstract

In this paper, using the notions of qκ2-quantum integral and qκ2-quantum derivative, we present some new identities that enable us to obtain new quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions. This paper, in particular, generalizes and expands previous findings in the field of quantum and classical integral inequalities obtained by various authors.

Some New Generalized Fractional Newton’s Type Inequalities for Convex Functions

“…These results can be helpful in finding the error bounds of Newton formulas in fractional calculus, which is the main motivation of this paper. Moreover, the main difference between the results proved in [11][12][13] and the results of this paper is that while the papers [11,12] are derived on Newton type inequalities for quantum integrals and the paper [13] focus on Newton type inequalities for Riemann-Liouville fractional integrals operators, we prove some inequalities of Newton type by using the generalized fractional integrals. These inequalities generalize the results of the paper [13] and give some new inequalities for k-fractional integrals, Hadamard fractional integrals, conformable fractional integrals, etc.…”

“…Some bounds for the q-Simpson's and Newton's type inequalities were proved by Budak et al in [11]. Siricharuanun et al proved some inequalities of Simpson and Newton type by using quantum numbers in [12]. Until recent years, Newton-type inequalities for fractional integrals had not been proven.…”

On Some New Simpson’s Formula Type Inequalities for Convex Functions in Post-Quantum Calculus

“…Recently, Siricharuanun et al [29] proved the following Simpson's formula type inequality for convex functions.…”

“…In [24], Khan et al used the green function to prove quantum HH inequality. For convex and coordinated convex functions, the authors of [25][26][27][28][29][30] constructed new quantum Simpson's and quantum Newton's type inequalities. Consult [31][32][33] for quantum Ostrowski's inequality for convex and co-ordinated convex functions.…”

Some New Simpson’s-Formula-Type Inequalities for Twice-Differentiable Convex Functions via Generalized Fractional Operators

“…Thereupon, several mathematicians studied fractional Simpson inequalities for these kinds of fractional integral operators [11][12][13][14][15][16][17][18][19]. For more studies related to different integral operator inequalities, one can see [20][21][22][23][24][25][26][27][28][29][30][31]. In addition, Sarikaya et al obtained several Simpson-type inequalities for mappings whose second derivatives are convex [32].…”