Hermite-Hadamard and Simpson-Like Type Inequalities for Differentiable Harmonically Convex Functions

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 types for functions whose derivatives in absolute value at certain power are harmonically convex. Some applications to special means of real numbers are also given. 1. Introduction Let be a convex function defined on the interval of real numbers and with . 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) for appropriate particular selections of the mapping . Both inequalities hold in the reversed direction if is concave. The following inequality is well known in the literature as Simpson inequality. Theorem 1. Let be a four times continuously differentiable mapping on and . Then the following inequality holds: For some results which generalize, improve, and extend the Hermite-Hadamard and Simpson inequalities, one refers the reader to the recent papers (see [1–8]). In [9], the author introduced the concept of harmonically convex functions and established some results connected with the right-hand side of new inequalities similar to inequality (1) for these classes of functions. Some applications to special means of positive real numbers were also given. Definition 2. Let be a real interval. A function is said to be harmonically convex, if for all and . If inequality in (3) is reversed, then is said to be harmonically concave. The following result of the Hermite-Hadamard type holds. Theorem 3. Let be a harmonically convex function and with . If then the following inequalities hold The above inequalities are sharp. Some results connected with the right part of (4) were given in [9] as follows. Theorem 4. Let be a differentiable function on , with , and . If is harmonically convex on for , then where Theorem 5. Let be a differentiable function on , with , and . If is harmonically convex on for , , then where In this paper, one gives some general integral inequalities connected with the left and right parts of (4); as a result of this, one obtains some new midpoint, trapezoid, and Simpson-like type inequalities for differentiable harmonically convex functions. 2. Main Results In order to prove our main results we need the following lemma. Lemma 6. Let be a differentiable function on and with . If then for one has the equality where . Proof. It suffices to note that Set and , which gives