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

In this paper, we have established new weighted Hermite-Hadamard type inequalities for strongly GA-convex functions. Those findings are obtained by using geometric symmetry of continuous positive mappings and differentiable mappings whose derivative in absolute value are strongly GA-convex. Some previous results are special cases of the results obtained in this paper.


Introduction
introduced strongly convex functions. However, we can find some references (Merentes and Nikodem, 2010;Nikodem and Páles, 2011) citing Polyak (1966) as being the pioneer to introduce this notion. Karamardian (1969b) investigated the class of scalar functions whose gradients are strongly monotone. It is well known that every continuously differentiable function is strongly monotone if its Jacobian matrix is strongly positive definite (Karamardian, 1969a). Niculescu (2000) investigated the class of multiplicatively convex functions by replacing the arithmetic mean to the geometric mean. It is well known that every polynomial ( ) with nonnegative coefficients is a multiplicatively convex function on [0, ∞). More generally, every real analytic function ( ) = ∑ ∞ =0 with nonnegative coefficients is a multiplicatively convex function on (0, ), where denotes the radius of convergence (Niculescu, 2000). Niculescu (2000) showed that a continuous function : ⊂ (0, ∞) → [0, ∞) is multiplicatively convex if and only if , ∈ ⇒ (√ ) ≤ √ ( ) ( ). Qi and Xi (2014) introduced a new concept of geometrically quasi-convex functions and established some integral inequalities of Hermite-Hadamard type for the function whose derivatives are of geometric quasi-convexity (Qi and Xi, 2014). Noor et al. (2017) introduced generalized geometrically convex functions and derived some basic inequalities related to generalize geometrically convex functions. Noor et al. (2017) also established new Hermite-Hadamard type inequalities for generalized geometrically convex functions. 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.

Preliminaries
Let : [ , ] → ℝ be a convex function with < . Then the following double inequality is known as Hermite-Hadamard inequality in the literature.

Main results
In this section, we will discuss our main results.

Conclusion
In this paper, some new weighted Hermite-Hadamard type inequalities for strongly GA-convex functions are obtained by using geometric symmetry of a continuous positive mapping and a differentiable mapping whose derivatives in absolute value are strongly GA-convex.

Funding
The research of the first author is supported by the UGC-BHU Research Fellowship, through sanction letter no: Ref No./Math/Res/Sept. 2017/117 and the second author is financially supported by the Department of Science and Technology, SERB, New Delhi, India, through grant no. MTR/2018/000121.