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: where , , and are defined as in Theorem 7.
Proof. The assertion follows from inequality (14) in Theorem 7, for , .
Proposition 26. 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 27. 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 , .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.