Hermite–Hadamard type inequalities for n-times differentiable and geometrically quasi-convex functions

By Hölder’s integral inequality, the authors establish some Hermite–Hadamard type integral inequalities for n-times differentiable and geometrically quasi-convex functions.

(1) Theorem 2 (Xi and Qi 2013) Let f : I ⊆ R + → R be a differentiable function on I • and a, b ∈ I • with a < b. If |f ′ | is geometrically convex on [a, b], then where for u, v > 0 and u � = v is called the logarithmic mean.
Theorem 3 (Dragomir and Agarwal 1998)  Corresponding to the concept of geometrically convex functions, the geometrically quasi-convex functions were introduced in Qi and Xi (2014) as follows.
In Qi and Xi (2014), some integral inequalities of Hermite-Hadamard type for geometrically quasi-convex functions were established.
The aim of this paper is to find more integral inequalities of Hermite-Hadamard type for n-times differentiable and geometrically quasi-convex functions.

A Lemma
In order to obtain our main results, we need the following Lemma.
Lemma 1 (Wang and Shi 2016) For n ∈ N, let f : I ⊆ R + → R be a n-times differentiable function on Remark 1 Under the conditions of Lemma 1, taking n = 1, we obtain which can be found in Zhang et al. (2013).

Inequalities for geometrically quasi-convex functions
Now we start out to establish some new Hermite-Hadamard type inequalities for n-times differentiable and geometrically quasi-convex functions.
Theorem 5 For n ∈ N, suppose that f : I ⊆ R + → R is a n-times differentiable function on I • , that f (n) ∈ L 1 ([a, b]), and that a, b ∈ I with a < b. If f (n) q is geometrically quasi-convex on [a, b] for q ≥ 1, then Proof By the geometric quasi-convexity of f (n) q , Lemma 1, and Hölder's inequality, one has Theorem 5 is thus proved.

Corollary 1 Under the assumptions of Theorem 5, if q = 1, then
Theorem 6 For n ∈ N, suppose that f : I ⊆ R + → R is a n-times differentiable function on I • , that f (n) ∈ L 1 ([a, b]), and that a, b ∈ I with a < b. If f (n) q is geometrically quasi-convex on [a, b] for q > 1, then for 0 ≤ m, r ≤ (n + 1)q.
Proof From the geometric quasi-convexity of f (n) q , Lemma 1, and Hölder's inequality, we have The proof of Theorem 6 is complete.