Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in q-Calculus

Abstract

The main objective of this study is to establish two important right q-integral equalities involving a right-quantum derivative with parameter m∈[0,1]. Then, utilizing these equalities, we derive some new variants for midpoint- and trapezoid-type inequalities for the right-quantum integral via differentiable (α,m)-convex functions. The fundamental benefit of these inequalities is that they may be transformed into q-midpoint- and q-trapezoid-type inequalities for convex functions, classical midpoint inequalities for convex functions and classical trapezoid-type inequalities for convex functions are transformed without having to prove each one independently. In addition, we present some applications of our results to special means of positive real numbers. It is expected that the ideas and techniques may stimulate further research in this field.

On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities

“…Some improvements and extensions of the above result have been obtained by some authors, see, e.g., [4][5][6][7]. Other extensions of (1) to various classes of functions have been obtained: s-convex functions [8][9][10][11], log-convex functions [12][13][14], h-convex functions [15,16], and m-convex functions [17][18][19][20].…”

New Estimates for Hermite-Hadamard Inequality in Quantum Calculus via (α, m) Convexity

“…In recent years, many papers have been devoted to inequalities for quantum integrals. For some of them, one can refer to [23][24][25][26][27][28][29][30].…”

Error estimates of Hermite‐Hadamard type inequalities with respect to a monotonically increasing function

“…Omrani et al [16] created a Hermite‐Hadamard inequality for $$$ \left(p,h\right) $$$‐convex functions and also introduced this class. Zhao et al [17] inaugurated two crucial right $$$ q $$$‐integral equalities containing a right‐quantum derivative with parameter $$$ m\in \left[0,1\right] $$$ and then derives some novel variants for midpoint and trapezoid‐type inequalities for the right‐quantum integral via differentiable $$$ \left(\alpha, m\right) $$$‐convex functions. Agarwal et al [18–20] explored new Ostrowski type and Hermite‐Hadamard type inequalities based on the concept of different convexities and presented their applications to the special means and midpoint formulae.…”