Mathematics and Computer Science
Volume 1, Issue 4, November 2016, Pages: 86-92

Hermite-Hadamard Type Integral Inequalities for Log-η-Convex Functions

Mohsen Rostamian Delavar1, *, Farhad Sajadian2

1Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran

2Department of Mathematics, Semnan University, Semnan, Iran

Email address:

(M. R. Delavar)
(F. Sajadian)

*Corresponding author

To cite this article:

Mohsen Rostamian Delavar, Farhad Sajadian. Hermite-Hadamard Type Integral Inequalities for Log--Convex Functions. Mathematics and Computer Science. Vol. 1, No. 4, 2016, pp. 86-92. doi: 10.11648/j.mcs.20160104.13

Received: September 9, 2016; Accepted: October17, 2016; Published:November 9, 2016


Abstract: In this paper by using the concept of log-η-convexity of functions some interesting inequalities are investigated. In fact new Hermite-Hadamard type integral inequalities involving log-η-convex function are established. The obtained results have as particular cases those previously obtained for log-convex.

Keywords: Log-η-Convex Functions, Integral Inequalities, Hermite-Hadamard Type Inequalities


1. Introduction and Preliminaries

The elegance in shape and interesting properties of convex functions make it attractive to study this class of function in mathematical analysis specially in applied mathematical analysis. In the last 60 years many efforts have gone on generalization of notion of convexity. In our opinion the following classification in generalization of convex functions holds:

(1) Works that change the form of defining convex functions to a generalized form such as quasi-convex [5], pseudo-convex [16], strongly convex [19], logarithmically convex [18], approximately convex [11], delta-convex[20], h-convex [22], midconvex functions [13], etc.

(2) Works that extend the domain set of convex functions such as E-convex functions [23], -convex functions, all works on convex functions from  to R [3], invex functions [10], etc.

(3) Works that extend the range set of convex functions such as works on functions with range in vector spaces [12], all kind of multivalued convex functions [2,14], etc.

On the other hand logarithmically convex (log-convex) functions are interesting class of functions to study in many fields of mathematics. They have been found to play an important role in the theory of special functions and mathematical statistics. To see recent works about log-convex functions see [4,17,18]).

Motivated by above works, we use the concept of log--convex function to establish some new Hermite-Hadamard type integral inequalities involving log--convex function. In fact obtained results have as particular cases those previously obtained for log-convex. We start with two definitions and one example.

Let  be an interval in real line . Consider  for appropriate .

Definition 1. [4] A function  is called convex with respect to  (briefly -convex), if

(1)

for all  and .

In fact above definition geometrically says that if a function is -convex on , then its graph between any  is on or under the path starting from  and ending at . If  should be the end point of the path for every , then we have  and the function reduces to a convex one.

Definition 2. Consider  and . If

(2)

for every  and , then  is called log--convex function.

In the above definition if we set , then we recapture the classic definition of a log-convex function. It is clear that  is log--convex iff  is -convex and when  is -convex then  is log--convex.

The following are two simple examples of log--convex functions.

Example 1.

a. Consider a function  defined by

and define a bifunction  as , for all  It is not hard to check that  is a log--convex function.

b. Define the function  by

and define the bifunction  by

Then  is log--convex.

The following result is of importance [3]:

Theorem 1. Suppose that  is a -convex function and  is bounded from above on . Then  satisfies a Lipschitz condition on any closed interval  contained in the interior  of . Hence,  is absolutely continuous on  and continuous on .

Note. As a consequence of Theorem 1, if  is a log--convex function where  is bounded from above on , then  is integrable and so  is integrable. For other results see [2, 4].

Some Hermite-Hadamard type inequalities related to -convex functions are proved in [3, 4, 7]. Some log--convex version of this type inequalities are investigated in the following.

provided that  is a -convex function,  is bounded from above on  and  is upper bound of .

Now if  is log--convex, since  is -convex we have

Consequently

So

Also if we consider

(a) (Arithmetic mean) , for any ,

(b) (Geometric mean) , for any ,

then we have

(3)

For more results about this inequalities see [2,5].

The following theorem is a consequence of Theorem  of [1], which we use these results frequently in this paper.

Theorem 2. If  and  are positive increasing functions on . Then

Also if  and  are positive decreasing functions on  and  is an upper bound for  and , then  and  are positive increasing functions and we have

which gives again

2. Main Results

In this section by using log--convexity property of a function some inequalities which generalize those previously obtained for log-convex functions are given.

Theorem 3. Let  be a log--convex function with  bounded from above on  and  be the upper bound of the function .

(4)

Consider  with . Then

 

Proof. For any  and ,

and

Then for

and

So

or

Now choosing  and      for all  we get

(5)

Now the left side of (4) is a consequence of (5) with integration over .

For the right side of (4), using the elementary inequality   and relations

we get

With the same argument we can obtain that

So

where for the last inequality we used the property that

When a log--convex function is positive and increasing, we can use Theorem 2 to obtain the following inequalities as well.

Theorem 4. Let  be an increasing log--convex function with  bounded from above on . Also consider ,  and . Then

Proof. For any  we have

and for every  we have [6],

So we can write

Then

Also

Therefor we have

On the other hand

Hence

which gives

(6)

The following result is obtained for the multiplication of two positive increasing log--convex functions under some special conditions.

Theorem 5. Let  be increasing log--convex functions with  bounded from above on . Also consider  with , ,  and . Then the following inequality holds:

where  and .

Proof. Since  are log--convex functions, we have

for all . So

Therefor one can write:

(7)

Integration from (7) over  on  gives

On the other hand

and

There for

The dual form of Theorem 5, is stated as the following.

Theorem 6. Let  be increasing log--convex functions with  bounded from above on . Also consider  with , ,  and . Then the following inequality holds:

where  and .

Proof. Change the role of  and  in proof of Theorem 5.

Using an elementary inequality between real numbers leads to an inequality related to square of a positive increasing log--convex function.

Theorem 7. Let  be an increasing log--convex function with  bounded from above on . Also consider  with  and . Then

Proof. Since  is log--convex function on , we have

for all . Using the elementary inequality

and the fact that  is bounded from above we have

(8)

Then by integration over  in (8),

(9)

It is easy to check the following from (9):

So

3. Conclusion

Logarithmically convex (log-convex) functions have some nice results in mathematical inequalities and are of interest in many areas of mathematics. They play a valuable and important role in the theory of special functions and mathematical statistics. On the other hand it should be noticed that in new problems related to convexity, generalized notions about convexity are required to obtain applicable results. One of these generalizations may be notion of log--convex functions which results in many interesting integral inequalities such as generalized form of Hermite-Hadamard type integral inequalities.


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