Refinements of Hermite-Hadamard Inequalities for Functions When a Power of the Absolute Value of the Second Derivative Is P-Convex
A. Barani,1S. Barani,2and S. S. Dragomir3,4
Academic Editor: Renat Zhdanov
Received10 Mar 2012
Accepted10 May 2012
Published05 Jul 2012
Abstract
We extend some estimates of the right-hand side of Hermite-Hadamard-type inequalities for functions whose second derivatives absolute values are -convex. Applications to some special means are considered.
1. Introduction
Let be a convex function defined on the interval of real numbers and with . The following double inequality
is known in the literature as the Hermite-Hadamard inequality. Both inequalities hold in the reversed direction if is concave. We note that Hermite-Hadamard inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensen's inequality. Note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the function . Both inequalities hold in the reversed direction if is concave (see [1]).
It is well known that the Hermite-Hadamard inequality plays an important role in nonlinear analysis. Over the last decade, this classical inequality has been improved and generalized in a number of ways; there have been a large number of research papers written on this subject, (see, [2β13]) and the references therein. In [13] Dragomir and Agarwal established the following results connected with the right-hand side of (1.1) as well as applied them for some elementary inequalities for real numbers and numerical integration.
Theorem 1.1. Assume that with and is a differentiable function on . If is convex on , then the following inequality holds:
Theorem 1.2. Assume that with and is a differentiable function on . Assume with . If is convex on , then the following inequality holds:
In [1] Pearce and PeΔariΔ proved the following theorem.
Theorem 1.3. Let be a differentiable function on with . If is convex on , for , then the following inequality holds:
Recall that the function is said to be quasiconvex if for every we have
The generalizations of the Theorems 1.1 and 1.2 are introduced by Ion in [14] for quasiconvex functions and are given in [6] to differentiable -convex functions. Then, Alomari et al. in [2] improved the results in [14] and Theorem 1.3, for twice differentiable quasiconvex functions.
On the other hand, Dragomir et al. in [11] defined the following class of functions.
Definition 1.4. Let be an interval. The function is said to be -convex (or belong to the class ) if it is nonnegative and, for all and , satisfies the inequality
Note that contain all nonnegative convex and quasiconvex functions. Since then numerous articles have appeared in the literature reflecting further applications in this category, see [3, 6, 12, 15, 16] and references therein. Ozdemir and Yildiz in [15] proved the following results.
Theorem 1.5. Let be a twice differentiable function on and with . If is -convex, , then the following inequality holds:
Corollary 1.6. If in Theorem 1.5 one chooses , one obtains
Theorem 1.7. Let be a twice differentiable function on and with . If is -convex, and , then the following inequality holds:
Corollary 1.8. If in Theorem 1.7 one chooses , one obtains
The main purpose of this paper is to establish the refinements of results in [15]. Applications for special means are considered.
2. Main Results
In order to prove our main theorems, we need the following Lemma in [5] throughout this paper.
Lemma 2.1. Suppose that is a twice differentiable function on , the interior of . Assume that , with and , is integrable on . Then, the following equality holds:
In the following theorem, we will propose some new upper bound for the right-hand side of (1.1) for -convex functions, which is better than the inequality had done in [15].
Theorem 2.2. Let be a twice differentiable function on such that is a -convex function on . Suppose that with and . Then, the following inequality holds:
Proof. Since is a -convex function, by using Lemma 2.1 we get
An immediate consequence of Theorem 2.2 is as follows.
Corollary 2.3. Let be as in Theorem 2.2, if in addition
(i) , then one has
(ii) , then one has
The corresponding version for powers of the absolute value of the second derivative is incorporated in the following theorem.
Theorem 2.4. Let be a differentiable function on . Assume that , such that is a -convex function on . Suppose that with and . Then, the following inequality holds:
where .
Proof. By assumption, Lemma 2.1 and HΓΆlder's inequality, we have
where . We note that the Beta and Gamma functions are defined, respectively, as follows
and are used to evaluate the integral . Indeed, by setting , we get
and using property
of Beta function, we obtain
where and the proof is completed.
The following corollary is an immediate consequence of Theorem 2.4.
Corollary 2.5. Let be as in Theorem 2.4, if in addition
(i) , then one has
(ii) , then one has
Another similar result may be extended in the following theorem.
Theorem 2.6. Let be a differentiable function on . Assume that such that is a -convex function on . Suppose that with and . Then, the following inequality holds:
Proof. Suppose that . From Lemma 2.1 and using well-known power mean inequality, we get
which completes the proof.
Corollary 2.7. Let be as in Theorem 2.6, if in addition(i), then (2.4) holds,(ii), (2.5) holds.
3. Applications to Special Means
Now, we consider the applications of our theorems to the special means. We consider the means for arbitrary real numbers , (). We take the following
(1) Arithmetic mean:
(2) Logarithmic mean:
(3) Generalized log-mean:
Now, using the results of Section 2, we give some applications for special means of real numbers.
Proposition 3.1. Let , , and , . Then, one has
Proof. The assertion follows from Theorem 2.2 applied to the -convex function , .
Proposition 3.2. Let , , and . Then, for all one has
where .
Proof. The assertion follows from Theorem 2.4 applied to the -convex function , .
Proposition 3.3. Let , , and , . Then, for all one has
Proof. The assertion follows from Theorem 2.6 applied to the -convex function , .
References
C. E. M. Pearce and J. Pečarić, βInequalities for differentiable mappings with application to special means and quadrature formula,β Applied Mathematics Letters, vol. 13, no. 2, pp. 51β55, 2000.
M. Alomari, M. Darus, and S. S. Dragomir, βNew inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex,β Tamkang Journal of Mathematics, vol. 41, no. 4, pp. 353β359, 2010.
A. O. Akdemir and M. E. Ozdemir, βSome Hadamard-type inequalities for coordinated -convex functions and Godunova-Levin functions,β in Proceedings of the AIP Conference Proceedings, vol. 1309, pp. 7β15, 2010.
A. Barani, A. G. Ghazanfari, and S. S. Dragomir, βHermite-Hadamard inequality through prequasiinvex functions,β RGMIA Research Report Collection, vol. 14, article 48, 2011.
A. Barani, S. Barani, and S. S. Dragomir, βRefinements of Hermite-Hadamard type inequality for functions whose second derivatives absolute values are quasiconvex,β RGMIA Research Report Collection, vol. 14, article 69, 2011.
A. Barani and S. Barani, βHermite-Hadamard inequality for functions whose derivatives absolute values are P-convex,β Bulletin of the Australian Mathematical Society. In press.
S. S. Dragomir and C. E. M. Pearce, βSelected Topics on Hermite-Hadamard Inequalities and Applications,β RGMIA Monographs, Victoria University, 2000, http://ajmaa.org/RGMIA/monographs.php/.
S. S. Dragomir, βTwo mappings in connection to Hadamard's inequalities,β Journal of Mathematical Analysis and Applications, vol. 167, no. 1, pp. 49β56, 1992.
S. S. Dragomir, J. E. Pečarić, and J. Sándor, βA note on the Jensen-Hadamard inequality,β L'Analyse Numérique et la Théorie de l'Approximation, vol. 19, no. 1, pp. 29β34, 1990.
S. S. Dragomir, J. Pečarić, and L. E. Persson, βSome inequalities of Hadamard type,β Soochow Journal of Mathematics, vol. 21, no. 3, pp. 335β341, 1995.
S. S. Dragomir and C. E. M. Pearce, βQuasi-convex functions and Hadamard's inequality,β Bulletin of the Australian Mathematical Society, vol. 57, no. 3, pp. 377β385, 1998.
S. S. Dragomir and R. P. Agarwal, βTwo inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula,β Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 11, no. 5, pp. 91β95, 1998.
D. A. Ion, βSome estimates on the Hermite-Hadamard inequality through quasi-convex functions,β Analele Universit\u a\c tii din Craiova. Seria Matematic\u a-Informatic\u a. Annals of the University of Craiova. Mathematics and Computer Science Series, vol. 34, pp. 83β88, 2007.
K. L. Tseng, G. S. Yang, and S. S. Dragomir, βOn quasiconvex functions and Hadamard’s inequality,β RGMIA Research Report Collection, vol. 14, article 1, 2003.