Weighted version of Hermite–Hadamard type inequalities for geometrically quasi-convex functions and their applications

Abstract

This paper presents new weighted Hermite–Hadamard type inequalities for a new class of convex functions which are known as geometrically quasi-convex functions. Some applications of these results to special means of positive real numbers have also been presented. These findings have been proved to be useful for researchers working in the fields of numerical analysis and mathematical inequalities.

A weighted version of Hermite-Hadamard type inequalities for strongly GA-convex functions

“…This completes the proof. Remark 3: If | ′| is geometrically quasi convex, then the above theorem reduces to Theorem 3 of Obeidat and Latif (2018).…”

“…For more details, one can refer to (Latif, 2014;Niculescu and Persson, 2006;Noor et al, 2014a;2014b;Shuang et al, 2013;Zhang et al, 2013). Recently, Obeidat and Latif (2018) established some new weighted Hermite-Hadamard type inequalities for geometrically quasi-convex functions and also showed how we can use inequalities of Hermite-Hadamard type to obtain the inequalities for special means. For more details on Hermite-Hadamard inequalities, we refer the interested reader (Dragomir and Pearce, 2003;Latif, 2014;Shuang et al, 2013;Zhang et al, 2013;Qi et al, 2005).…”

“…Motivated by Noor et al (2017) and Obeidat and Latif (2018), we establish some new weighted Hermite-Hadamard inequalities for strongly GAconvex functions by using geometric symmetry of a continuous positive mapping and a differentiable mapping whose derivatives in absolute value are strongly GA-convex.…”

“…Definition 3: A function ψ: G ⊆ ℝ + = (0, ∞) → ℝ is said to be geometrically symmetric with respect to √cd if ψ ( cd x ) = ψ(x) for every x ∈ G (Obeidat and Latif, 2018).…”

“…If μ = 0 in Theorem 1, then, If |ψ′| is geometrically quasi-convex, then the above theorem reduces to Theorem 1 ofObeidat and Latif (2018). Let ψ: G ⊆ ℝ + = (0, ∞) → ℝ be a differentiable function on G 0 and c, d ∈ G 0 with c < d, and let λ: [c, d] → [0, ∞) be a continuous positive mapping and geometrically symmetric to √cd.…”