On q-Hermite–Hadamard inequalities for general convex functions

We first establish two new identities, based on the kernel functions with either two section or three sections, involving quantum integrals by using new definition of quantum derivative. Then, some new inequalities related to Simpson's 1/3 formula for convex mappings are provided. In addition, Newton type inequalities, for functions whose quantum derivatives in modulus or their powers are convex, are deduced. We also mention that the results in this work generalize inequalities given in earlier study.

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“…Definition 33 For a continuous function

μ : false[m, n false] \to double-struckR

, then q n ‐derivative of μ at x ∈[ m , n ] is characterized by the expression

^{n} D_{q} μ (x) = \frac{μ (q x + (1 - q) n) - μ (x)}{(1 - q) (n - x)}, x \neq n .

…”

Section: Preliminaries Of Q‐calculus and Some Inequalitiesmentioning

confidence: 99%

“…Definition 33 Let

μ : false[m, n false] \to double-struckR

be a continuous function. Then, the q n ‐definite integral on

([], m, n)

is defined as

\int_{m}^{n} μ (x)^{n} d_{q} x = (1 - q) (n - m) \sum_{k = 0}^{\infty} q^{k} μ (q^{k} m + (1 - q^{k}) n) .

…”

Section: Preliminaries Of Q‐calculus and Some Inequalitiesmentioning

confidence: 99%

Simpson and Newton type inequalities for convex functions via newly defined quantum integrals

Budak

Erden

Ali

2020

Math Methods in App Sciences

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“…On the other hand, Bermudo et al [22] gave the following definition and obtained the related Hermite-Hadamard-type inequalities.…”

Section: Preliminaries Of Q-calculus and Some Inequalitiesmentioning

confidence: 99%

Quantum Hermite–Hadamard-type inequalities for functions with convex absolute values of second $q^{b}$-derivatives

et al. 2021

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In this paper, we obtain Hermite–Hadamard-type inequalities of convex functions by applying the notion of $q^{b}$ q b -integral. We prove some new inequalities related with right-hand sides of $q^{b}$ q b -Hermite–Hadamard inequalities for differentiable functions with convex absolute values of second derivatives. The results presented in this paper are a unification and generalization of the comparable results in the literature on Hermite–Hadamard inequalities.

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“…For more detail about q b -integrals and corresponding inequalities, we refer to [8]. We have to give the following notation, which will be used many times in the next sections (see [34]):…”

Section: Definition 5 Letmentioning

confidence: 99%

“…In 2013, Tariboon [7] introduced the a D q -difference operator. Recently, in 2020, Bermudo et al [8] introduced the notions of the b D q -derivative and integral.…”

Section: Introductionmentioning

confidence: 99%

Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables

et al. 2021

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On q-Hermite–Hadamard inequalities for general convex functions

Cited by 148 publications

References 8 publications

Simpson and Newton type inequalities for convex functions via newly defined quantum integrals

Simpson and Newton type inequalities for convex functions via newly defined quantum integrals

Quantum Hermite–Hadamard-type inequalities for functions with convex absolute values of second $q^{b}$-derivatives

Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables

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