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.
KeywordsHermite-Hadamard’s inequality invex set preinvex function prequasiinvex 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 .
2 Main results
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.
3 Applications to special means
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.
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