Abstract
In this paper, based on (alpha,m)-convexity, we establish different type inequalities via quantum integrals. These inequalities generalize some results given in the literature.
Highlights
Introduction and preliminariesThroughout the paper, let I := [a, b] ⊆ R with 0 ≤ a < b be an interval, I◦ be the interior of I and let 0 < q < 1 be a constant.Let f : I → R be convex on I, the Hermite–Hadamard inequality holds:f a + b ≤ 1 b f (x) dx ≤ f (a) + f (b) . (1.1) b–a aIf f : I → R is four times continuously differentiable on I◦ and f (4) ∞ = supx∈(a,b) |f (4)(x)| < ∞, the Simpson inequality holds:1 f (a) + f (b) a+b+ 2f b–a b f (x) dx ≤a f (4) ∞(b – a)4. (1.2)Many researchers generalized the inequalities (1.1) and (1.2)
This paper aims to establish different types of quantum integral inequalities via (α, m)convexity
In the present research, based on a new quantum integral identity with multiple parameters, we have developed some quantum error estimations of different type inequalities through (α, m)-convexity, such as the midpoint-like inequalities, the Simpson-like inequalities, the averaged midpoint-trapezoid-like inequalities and the trapezoid-like inequalities
Summary
0dqt, which is presented by Sudsutad et al in [26, Lemma 3.1]. ∞ n=0 μτ +1q(τ +1)n μτ+1(1 – q) 1 – qτ+1 and (1 – t)τ 0dqt = (1 – q)μ qn 1 – qnμ τ. Proof When (λ + q)μ ≤ λ, making use of Lemma 2.2, we get μ μ tτ qt – (λ – λμ) 0dqt = (λ – λμ)tτ – qtτ+1 0dqt μτ+1(1 – q)(λ – λμ) qμτ+2(1 – q). When (λ + q)μ > λ, making use of Lemma 2.2 again, we get μ tτ qt – (λ – λμ) 0dqt λ–λμ q (λ – λμ)tτ – qtτ+1 0dqt +. Theorem 3.1 Let f : I → R be a convex function on [a, b] with 0 < q < 1. Using Lemma 2.1, we can obtain the following theorem
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