Research Article | Open Access

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

**Academic Editor:**Roberto A. Kraenkel

#### Abstract

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
Similarly, we can show that
Thus,
which is required.

Theorem 7. *Let be a differentiable function on , with , and . If is harmonically convex on for and then one has the following inequality for :
**
where
*

*Proof. *Let . From Lemma 6 and using the Hölder inequality, we have
Hence, by harmonically convexity of on , we have
It is easily to check that
This concludes the proof.

Corollary 8. *Under the assumptions of Theorem 7 with , one has
**
where
*

Corollary 9. *Under the assumptions of Theorem 7 with , one has
**
where
*

Corollary 10. *Under the assumptions of Theorem 7 with , one has
**
where
*

Theorem 11. *Let be a differentiable function on , with , and . If is harmonically convex on for and then one has the following inequality for **
where
**
and .*

*Proof. *Let . Using Lemma 6 and Hölder’s integral inequality, we deduce
Using the harmonically convexity of , we obtain

Further, we have
A combination of (27)–(29) gives the required inequality (25).

Corollary 12. *Under the assumptions of Theorem 11 with , one has
*

Corollary 13. *Under the assumptions of Theorem 11 with , one has
*

Corollary 14. *Under the assumptions of Theorem 11 with , one has
*

Theorem 15. *Let be a differentiable function on , with , and . If is harmonically convex on for and then one has the following inequality for :
**
where
**
and .*

*Proof. *Let . Using Lemma 6 and Hölder’s integral inequality, we deduce
Using the harmonically convexity of , we obtain
Further, we have
A combination of (35)–(37) gives the required inequality (33).

Corollary 16. *Under the assumptions of Theorem 15 with , one has
*

Corollary 17. *Under the assumptions of Theorem 15 with , one has
*

Corollary 18. *Under the assumptions of Theorem 15 with , one has
**
where
*

#### 3. Some Applications for Special Means

Let us recall the following special means of two nonnegative number with .(1)The arithmetic mean (2)The geometric mean (3)The harmonic mean (4)The logarithmic mean (5)The -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 is monotonically increasing over , denoting and .

Proposition 19. *Let and . Then one has the following inequality:
**
where is defined as in Theorem 7.*

*Proof. *The assertion follows from inequality (14) in Theorem 7, for .

Proposition 20. *Let and . Then one has the following inequality:
**
where , , and , , and are defined as in Theorem 11.*

*Proof. *The assertion follows from inequality (25) in Theorem 11, for .

Proposition 21. *Let and . Then one has the following inequality:
**
where , , and and are defined as in Theorem 15.*

*Proof. *The assertion follows from inequality (33) in Theorem 15, for .

Proposition 22. *Let , and . Then one has the following inequality:
**
where , and are defined as in Theorem 7.*

*Proof. *The assertion follows from inequality (14) in Theorem 7, for .

Proposition 23. *Let and . Then one has the following inequality:
**
where , , and , , and are defined as in Theorem 11.*

*Proof. *The assertion follows from inequality (25) in Theorem 11, for .

Proposition 24. *Let and . Then one has the following inequality:
**
where , , and , , , and are defined as in Theorem 15.*

*Proof. *The assertion follows from inequality (33) in Theorem 15, for .

Proposition 25. *Let , , , and . Then one has the following inequality:
*