Hermite-Hadamard Type Integral Inequalities for Log-η-Convex Functions
Mohsen Rostamian Delavar^{1, *}, Farhad Sajadian^{2}
^{1}Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran
^{2}Department of Mathematics, Semnan University, Semnan, Iran
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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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