- Open Access
Some weighted integral inequalities for differentiable preinvex and prequasiinvex functions with applications
© Latif and Dragomir; licensee Springer. 2013
- Received: 7 August 2013
- Accepted: 12 November 2013
- Published: 5 December 2013
In this paper, we present weighted integral inequalities of Hermite-Hadamard type for differentiable preinvex and prequasiinvex functions. Our results, on the one hand, give a weighted generalization of recent results for preinvex functions and, on the other hand, extend several results connected with the Hermite-Hadamard type integral inequalities. Applications of the obtained results are provided as well.
MSC: 26D15, 26D20, 26D07.
- Hermite-Hadamard’s inequality
- invex set
- preinvex function
- Hölder’s integral inequality
- power-mean inequality
Both the inequalities in (1.1) hold in reversed direction if f is concave. Inequalities (1.1) are famous in mathematical literature due to their rich geometrical significance and applications and are known as the Hermite-Hadamard inequalities (see ).
In , Dragomir and Agarwal obtained the following inequalities for differentiable functions which estimate the difference between the middle and the rightmost terms in (1.1).
Theorem 1 
Theorem 2 
where and .
Theorem 3 
for all and . Clearly, any convex function is a quasi-convex function. Furthermore, there exist quasi-convex functions which are not convex (see ).
Recently, Ion  introduced two inequalities of the right-hand side of Hadamard type for quasi-convex functions, as follows.
Theorem 4 
Theorem 5 
In , Alomari et al. established Hermite-Hadamard-type inequalities for quasi-convex functions which give refinements of those given above in Theorem 4 and Theorem 5.
Theorem 6 
Theorem 7 
Theorem 8 
In , Hwang established the following results for convex and quasi-convex functions; those results provide a weighted generalization of the results given in Theorem 1, Theorem 3, Theorem 6 and Theorem 8.
Theorem 9 
where and .
Theorem 10 
where and are as defined in Theorem 9.
Theorem 11 
where and are as defined in Theorem 9.
Theorem 12 
where and are as defined in Theorem 9.
In recent years, a lot of efforts have been made by many mathematicians to generalize the classical convexity. These studies include, among others, the work of Hanson , Ben-Israel and Mond , Pini , Noor [22, 23], Yang and Li  and Weir and Mond . Ben-Israel and Mond , Weir and Mond  and Noor [22, 23] have studied the basic properties of the preinvex functions and their role in optimization, variational inequalities and equilibrium problems. Hanson  introduced invex functions as a significant generalization of the convex functions. Ben-Israel and Mond  gave the concept of preinvex functions which is a special case of invexity. Pini  introduced the concept of prequasiinvex functions as a generalization of invex functions.
Let us recall some known results concerning preinvexity and prequasiinvexity.
K is said to be an invex set with respect to η if K is invex at each . The invex set K is also called an η-connected set.
Definition 1 
The function f is said to be preconcave if and only if −f is preinvex.
It is to be noted that every convex function is preinvex with respect to the map , but the converse is not true; see, for instance, .
Definition 2 
Also every quasi-convex function is prequasiinvex with respect to the map , but the converse does not hold; see, for example, .
In the recent paper, Noor  obtained the following Hermite-Hadamard inequalities for the preinvex functions.
Theorem 13 
Barani et al. in  presented the following estimates of the right-hand side of a Hermite-Hadamard-type inequality in which some preinvex functions are involved.
Theorem 14 
Theorem 15 
In , Barani et al. gave similar results for prequasiinvex functions as follows.
Theorem 16 
Theorem 17 
Latif  proved the following results which give a refinement of the results given in Theorems 14-17.
Theorem 18 
Theorem 19 
Theorem 20 
In the present paper we give new inequalities of Hermite-Hadamard for functions whose derivatives in absolute value are preinvex and prequasiinvex. Our results extend those results presented in very recent results from [2, 3, 5, 6] and  and generalize those results from [29, 30] and .
The following lemma is essential in establishing our main results in this section.
which is the required result. □
Remark 1 If we take , then Lemma 1 reduces to Lemma 2.1 from .
Now using Lemma 1, we shall propose some new upper bounds for the difference between the rightmost and middle terms of a weighted version of the Hadamard inequality (1.15) using preinvex and prequasiinvex mappings. Our results provide a weighted generalization of those results given in [29, 30] and .
In what follows we use the notations and .
Using (2.9) in (2.8), we get the required inequality. This completes the proof of the theorem. □
Remark 2 In Theorem 21, if we take for all , then (2.4) becomes inequality (1.16).
Remark 3 If in Theorem 21, then (2.4) reduces to inequality (1.11) from .
Using the last inequality (2.12) in (2.11), we get the desired inequality. This completes the proof of the theorem as well. □
Remark 4 In Theorem 22 if we take for all with , then (2.10) reduces to inequality (1.17).
where , , , .
A similar result may be stated as follows.
Utilizing inequality (2.16) in (2.15), we get inequality (2.14). This completes the proof of the theorem. □
Remark 6 If we take in Theorem 23, then the inequality reduces to inequality (1.12) from .
This reveals that inequality (2.17) is better than the one given by (1.17) in Theorem 15 from .
Now we give our results for prequasiinvex functions.
A combination of (2.8), (2.19) and (2.20) gives the required inequality (2.18). □
- (1)if is non-decreasing, then the following inequality holds:(2.21)
- (2)if is non-increasing, then the following inequality holds:(2.22)
Remark 8 
- (1)if is non-decreasing, then the following inequality holds:(2.24)
- (2)if is non-increasing, then the following inequality holds:(2.25)
Remark 9 If in Theorem 24, then (2.18) reduces to inequality (1.13) established in Theorem 11 from , and inequalities (2.24) and (2.25) recapture the related inequalities given in the corollary of Theorem 11.
Remark 10 If in Remark 8, then (2.23) becomes inequality (1.8) of Theorem 6 from , and inequalities (2.24) and (2.25) recapture the related inequalities of the corollary of Theorem 6.
A combination of (2.11), (2.27) and (2.28) gives us the required inequality (2.26). This completes the proof of the theorem. □
- (1)if is non-decreasing for , then the following inequality holds:(2.29)
- (2)if is non-increasing for , then the following inequality holds:(2.30)
Remark 11 
- (1)if is non-decreasing, then the following inequality holds:(2.32)
- (2)if is non-increasing, then the following inequality holds:(2.33)
Remark 12 If we take in Remark 11, then (2.31) becomes inequality (1.9) of Theorem 7 from , and inequalities (2.32) and (2.33) become the related inequalities given in the corollary of Theorem 7.
A combination of (2.15), (2.35) and (2.36) gives us the required inequality (2.34). This completes the proof of the theorem. □
- (1)if is non-decreasing for , then the following inequality holds:(2.37)
- (2)if is non-increasing for , then the following inequality holds:(2.38)
Remark 13 
if is non-decreasing, then inequality (2.24) holds,
if is non-increasing, then inequality (2.25) holds.
Remark 14 If in Theorem 26, then (2.34) reduces to inequality (1.14) established in Theorem 12 from , and inequalities (2.37) and (2.38) recapture the related inequalities established in the corollary of Theorem 12.
Remark 15 If in Remark 13, then (1.22) becomes inequality (1.10) of Theorem 8 from , and inequalities (2.24) and (2.25) recapture the related inequalities of the corollary of Theorem 8.
In what follows we give certain generalizations of some notions for a positive valued function of a positive variable.
Definition 3 
Homogeneity: for all ;
Monotonicity: If and , then ;
- (1)The arithmetic mean:
- (2)The geometric mean:
- (3)The harmonic mean:
- (4)The power mean:
- (5)The identric mean:
- (6)The logarithmic mean:
- (7)The generalized log-mean:
It is well known that is monotonic nondecreasing over , with and . In particular, we have the inequality .
Now, let a and b be positive real numbers such that . Consider the function , which is one of the above mentioned means, therefore one can obtain variant inequalities for these means as follows.
for , where , . Letting in (3.1), (3.2) and (3.3), we can get the required inequalities for a different weight function , and the details are left to the interested reader.
- Pečarić J, Proschan F, Tong YL: Convex Functions, Partial Ordering and Statistical Applications. Academic Press, New York; 1991.MATHGoogle Scholar
- Alomari M, Darus M, Kirmaci US: Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means. Comput. Math. Appl. 2010, 59: 225–232. 10.1016/j.camwa.2009.08.002MathSciNetView ArticleMATHGoogle Scholar
- Dragomir SS, Agarwal RP: Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula. Appl. Math. Lett. 1998, 11(5):91–95. 10.1016/S0893-9659(98)00086-XMathSciNetView ArticleMATHGoogle Scholar
- Dragomir SS: Two mappings in connection to Hadamard’s inequalities. J. Math. Anal. Appl. 1992, 167: 42–56.View ArticleMATHGoogle Scholar
- Hwang D-Y: Some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables. Appl. Math. Comput. 2011, 217(23):9598–9605. 10.1016/j.amc.2011.04.036MathSciNetView ArticleMATHGoogle Scholar
- Ion DA: Some estimates on the Hermite-Hadamard inequality through quasi-convex functions. An. Univ. Craiova, Math. Comput. Sci. Ser. 2007, 34: 82–87.MathSciNetGoogle Scholar
- Kırmacı US: Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 2004, 147(1):137–146. 10.1016/S0096-3003(02)00657-4MathSciNetView ArticleMATHGoogle Scholar
- Kırmacı US, Özdemir ME: On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 2004, 153(2):361–368. 10.1016/S0096-3003(03)00637-4MathSciNetView ArticleMATHGoogle Scholar
- Lee KC, Tseng KL: On a weighted generalization of Hadamard’s inequality for G -convex functions. Tamsui Oxford Univ. J. Math. Sci. 2000, 16(1):91–104.MathSciNetMATHGoogle Scholar
- Lupas A: A generalization of Hadamard’s inequality for convex functions. Univ. Beogr. Publ. Elektroteh. Fak., Ser. Mat. Fiz. 1976, 544–576: 115–121.MathSciNetGoogle Scholar
- Pearce CEM, Pečarić J: Inequalities for differentiable mappings with application to special means and quadrature formulae. Appl. Math. Lett. 2000, 13(2):51–55. 10.1016/S0893-9659(99)00164-0MathSciNetView ArticleMATHGoogle Scholar
- Qi F, Wei Z-L, Yang Q: Generalizations and refinements of Hermite-Hadamard’s inequality. Rocky Mt. J. Math. 2005, 35: 235–251. 10.1216/rmjm/1181069779MathSciNetView ArticleMATHGoogle Scholar
- Sarıkaya MZ, Aktan N: On the generalization some integral inequalities and their applications. Math. Comput. Model. 2011, 54(9–10):2175–2182. 10.1016/j.mcm.2011.05.026View ArticleMathSciNetMATHGoogle Scholar
- Sarikaya MZ, Avci M, Kavurmaci H: On some inequalities of Hermite-Hadamard type for convex functions. AIP Conf. Proc. 2010., 1309: Article ID 852. ICMS International Conference on Mathematical ScienceGoogle Scholar
- Sarikaya MZ: O new Hermite-Hadamard Fejér type integral inequalities. Stud. Univ. Babeş-Bolyai, Math. 2012, 57(3):377–386.MathSciNetMATHGoogle Scholar
- Saglam A, Sarikaya MZ, Yıldırım H: Some new inequalities of Hermite-Hadamard’s type. Kyungpook Math. J. 2010, 50: 399–410. 10.5666/KMJ.2010.50.3.399MathSciNetView ArticleMATHGoogle Scholar
- Wang C-L, Wang X-H: On an extension of Hadamard inequality for convex functions. Chin. Ann. Math. 1982, 3: 567–570.MATHGoogle Scholar
- Wu S-H: On the weighted generalization of the Hermite-Hadamard inequality and its applications. Rocky Mt. J. Math. 2009, 39(5):1741–1749. 10.1216/RMJ-2009-39-5-1741View ArticleMathSciNetMATHGoogle Scholar
- Hanson MA: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 1981, 80: 545–550. 10.1016/0022-247X(81)90123-2MathSciNetView ArticleMATHGoogle Scholar
- Ben-Israel A, Mond B: What is invexity? J. Aust. Math. Soc. Ser. B 1986, 28(1):1–9. 10.1017/S0334270000005142MathSciNetView ArticleMATHGoogle Scholar
- Pini R: Invexity and generalized convexity. Optimization 1991, 22: 513–525. 10.1080/02331939108843693MathSciNetView ArticleMATHGoogle Scholar
- Noor MA: Invex equilibrium problems. J. Math. Anal. Appl. 2005, 302: 463–475. 10.1016/j.jmaa.2004.08.014MathSciNetView ArticleMATHGoogle Scholar
- Noor MA: Variational-like inequalities. Optimization 1994, 30: 323–330. 10.1080/02331939408843995MathSciNetView ArticleMATHGoogle Scholar
- Yang XM, Li D: On properties of preinvex functions. J. Math. Anal. Appl. 2001, 256: 229–241. 10.1006/jmaa.2000.7310MathSciNetView ArticleMATHGoogle Scholar
- Weir T, Mond B: Preinvex functions in multiple objective optimization. J. Math. Anal. Appl. 1998, 136: 29–38.MathSciNetView ArticleMATHGoogle Scholar
- Noor MA: On Hadamard integral inequalities involving two log-preinvex functions. J. Inequal. Pure Appl. Math. 2007, 8(3):1–14.MathSciNetMATHGoogle Scholar
- Yang XM, Yang XQ, Teo KL: Characterizations and applications of prequasi-invex functions. J. Optim. Theory Appl. 2001, 110: 645–668. 10.1023/A:1017544513305MathSciNetView ArticleMATHGoogle Scholar
- Noor, MA: Hermite-Hadamard integral inequalities for log-preinvex functions. PreprintGoogle Scholar
- Barani A, Ghazanfari AG, Dragomir SS: Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex. RGMIA Res. Rep. Collect. 2011., 14: Article ID 64Google Scholar
- Barani A, Ghazanfari AG, Dragomir SS: Hermite-Hadamard inequality through prequasiinvex functions. RGMIA Res. Rep. Collect. 2011., 14: Article ID 48Google Scholar
- Latif MA: Some inequalities for differentiable prequasiinvex functions with applications. Konuralp J. Math. 2013, 1(2):17–29.MATHGoogle Scholar
- Sarikaya, MZ, Bozkurt, H, Alp, N: On Hermite-Hadamard type integral inequalities for preinvex and log-preinvex functions. arXiv:1203.4759v1
- Mohan SR, Neogy SK: On invex sets and preinvex functions. J. Math. Anal. Appl. 1995, 189: 901–908. 10.1006/jmaa.1995.1057MathSciNetView ArticleMATHGoogle Scholar
- Bullen PS: Handbook of Means and Their Inequalities. Kluwer Academic, Dordrecht; 2003.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.