Parametrized inequality of Hermite-Hadamard type for functions whose third derivative absolute values are quasi-convex

In this paper we present some inequalities of Hermite-Hadamard type for functions whose third derivative absolute values are quasi-convex. Moreover, an application to special means of real numbers is also considered.

Theorem 1.1 Let f : I ⊂ R → R be a differentiable mapping on I o , a, b ∈ I with a < b. If f ′ is quasi-convex on [a, b], then the following inequality holds: In 2010, Alomari et al. (2010a) established an analogous version of inequality (1), which is asserted by Theorem 1.2 below: Theorem 1.2 Let f : I ⊂ R → R be twice differentiable mapping on I o , a, b ∈ I with a < b and f ′′ is integrable on [a, b]. If f ′′ is quasi-convex on [a, b], then the following inequality holds: Recently, Guo et al. (2015) investigated Hermite-Hadamard type inequalities for geometrically quasi-convex functions. Qi (2014, 2015) and Xi et al. (2012 showed some new Hermite-Hadamard type inequalities for s-convex functions. For more results relating to refinements, counterparts, generalizations of Hadamard type inequalities, we refer interested readers to Alomari et al. (2010b), Chen (2015), Niculescu andPersson (2006), Pečarić et al. (1992), Sroysang (2014), Sroysang (2013) and Wu (2009).
The main purpose of this paper is to present a parametrized inequality of Hermite-Hadamard type for functions whose third derivative absolute values are quasi-convex. As applications, some new inequalities for special means of real numbers are established.

Lemmas
In order to prove our main results, we need the following lemmas.
Lemma 2.1 Let ǫ ∈ R and let f: ( Proof Integrating by parts, we have Changing variable x = a + (1 − )b, it follows that Thus, The proof of Lemma 2.1 is completed.

Lemma 2.2 Let ǫ be a real number. Then
Proof We distinguish three cases Our main results are stated in the following theorems.
Theorem 3.1 Let q ≥ 1 and ǫ ∈ R, and let f : I ⊂ R → R be three times differentiable on I • and a, b ∈ I with a < b. Assume that f ′′′ is integrable on [a, b], and f ′′′ q is quasiconvex on [a, b]. Then Proof Using Lemma 2.1 and Hölder's inequality gives By the quasi-convexity of f ′′′ q , we obtain Utilizing Lemma 2.2 leads to the desired inequality in Theorem 3.1.

Remark 3.2
It is worth noticing that if we use a substitution a → b, b → a and ǫ → 2 − ǫ in Theorem 3.1, we have the following further generalization of Theorem 3.1.
Theorem 3.3 Let q ≥ 1 and ǫ ∈ R, and let f : I ⊂ R → R be three times differentiable on I • , a, b ∈ I with a � = b. Assume that f ′′′ is integrable on [a, b], and f ′′′ q is quasi-convex on the closed interval formed by the points a and b. Then As a direct consequence, choosing ǫ = 1 in Theorem 3.3, we get the following inequality: Corollary 3.4 Let q ≥ 1 and ǫ ∈ R, and let f : I ⊂ R → R be three times differentiable on I • , a, b ∈ I with a � = b. Assume that f ′′′ is integrable on [a, b], and f ′′′ q is quasi-convex on the closed interval formed by the points a and b. Then In addition, if we utilize Theorem 3.1 with a substitution of ǫ = 0, 0.5, 3, −2, −3, −5, respectively, then we obtain the following results: Corollary 3.5 Let q ≥ 1 and ǫ ∈ R, and let f : I ⊂ R → R be three times differentiable on I • , a, b ∈ I with a � = b. Assume that f ′′′ is integrable on [a, b], and f ′′′ q is quasi-convex on the closed interval formed by the points a and b. Then