Hermite-Hadamard and Simpson-like type inequalities for differentiable harmonically convex functions

In this paper, a new identity for differentiable functions is derived. A consequence of the identity is that the author establishes some new general inequalities containing all of the Hermite-Hadamard and Simpson-like type for functions whose derivatives in absolute value at certain power are harmonically convex. Some applications to special means of real numbers are also given.


Introduction
Let f : I ⊂ R → R be a convex function defined on the interval I of real numbers and a, b ∈ I with a < b. The following inequality holds. This double inequality is known in the literature as Hermite-Hadamard integral inequality for convex functions. Note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the mapping f . Both inequalities hold in the reversed direction if f is concave.
Following inequality is well known in the literature as Simpson inequality: Theorem 1. Let f : [a, b] → R be a four times continuously differentiable mapping on (a, b) and f (4) ∞ = sup x∈(a,b) f (4) (x) < ∞. Then the following inequality holds: For some results which generalize, improve and extend the Hermite-Hadamard and Simpson inequalities, we refer the reader to the recent papers (see [1,2,3,4,6,7,8] ).
In [5], the author introduced the concept of harmonically convex functions and established some results connected with the right-hand side of new inequalities similar to the inequality (1.1) for these classes of functions. Some applications to special means of positive real numbers are also given. Definition 1. Let I ⊂ R\ {0} be a real interval. A function f : I → R is said to be harmonically convex, if for all x, y ∈ I and t ∈ [0, 1]. If the inequality in (1.2) is reversed, then f is said to be harmonically concave.
The following result of the Hermite-Hadamard type holds.
Theorem 2. Let f : I ⊂ R\ {0} → R be a harmonically convex function and a, b ∈ I with a < b. If f ∈ L[a, b] then the following inequalities hold The above inequalities are sharp.
Some results connected with the right part of (1.3) was given in [5] as follows: In this paper, we shall give some general integral inequalities connected with the left and right parts of (1.3), as a result of this, we shall obtained some new midpoint, trapezoid and Simpson like-type inequalities for differentiable harmonically convex functions.

Main results
In order to prove our main resuls we need the following lemma: Proof. It suffices to note that Setting x = ab At and dx = −ab(b−a) A 2 t dt, which gives Similarly, we can show that Thus, which is required.
for q ≥ 1 and then we have the following inequality for λ ∈ [0, 1] and From Lemma 1 and using the power mean inequality, we have It is easily check that This concludes the proof.

Some applications for special means
Let us recall the following special means of two nonnegative number a, b with b > a : (1) The arithmetic mean (2) The geometric mean (3) The harmonic mean (4) The Logarithmic mean

12İMDATİŞCAN
(5) The p-Logarithmic mean (6) the Identric mean These means are often used in numerical approximation and in other areas. However, the following simple relationships are known in the literature: It is also known that L p is monotonically increasing over p ∈ R, denoting L 0 = I and L −1 = L. Proposition 1. Let 0 < a < b and λ ∈ [0, 1]. Then we have the following inequality where C 1 is defined as in Theorem 5.
Proof. The assertion follows from the inequality (2.6) in Theorem 7, for f : Proposition 4. Let 0 < a < b, λ ∈ [0, 1] and q ≥ 1. Then we have the following inequality where C 1 , C 2 and C 3 are defined as in Theorem 5.