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Abstract Convexity and Hermite-Hadamard Type Inequalities
Journal of Inequalities and Applications volume 2009, Article number: 943534 (2009)
The deriving Hermite-Hadamard type inequalities for certain classes of abstract convex functions are considered totally, the inequalities derived for some of these classes before are summarized, new inequalities for others are obtained, and for one class of these functions the results on are generalized to . By considering a concrete area in , the derived inequalities are illustrated.
Studying Hermite-Hadamard type inequalities for some function classes has been very important in recent years. These inequalities which are well known for convex functions have been also found in different function classes (see [1–5]).
Abstract convex function is one of this type of function classes. Hermite-Hadamard type inequalities are studied for some important classes of abstract convex functions, and the concrete results are found [6–9]. For example, increasing convex-along-rays (ICAR) functions, which are defined in , are considered in . First, a correct inequality for these functions is given. For each there exists a number such that
Then, based on previous inequality the following inequality is proven. If , then for all continuous ICAR function
inequality is correct, where
and is the area of domain
In this article, first, the theorem which yields inequality (1.1) is proven for ICAR functions defined on , then the other inequalities are generalized, which based on this theorem.
Another generalization is made for . A more covering set is considered and all results (for IPH, ICR, InR, ICAR functions) are examined for this set.
2. Abstract Convexity and Hermite-Hadamard Type Inequalities
Let be a real line and . Consider a set and a set of functions defined on A function is called abstract convex with respect to (or -convex) if there exists a set such that
Clearly is -convex if and only if
Let be a set of functions A set is called a supremal generator of the set if each function is abstract convex with respect to
2.1. Increasing Positively Homogeneous Functions and Hermite-Hadamard Type Inequalities
A function defined on is called increasing (with respect to the coordinate-wise order relation) if implies that .
The function is positively homogeneous of degree one if
for all and
Let be the set of all min-type functions defined on
that is, the set consists of identical zero and all the functions of the form
One has that a function is convex if and only if is increasing and positively homogeneous of degree one (IPH) functions (see ).
The Hermite-Hadamard type inequalities are shown for IPH functions by using the following proposition which is very important for IPH functions.
Let be an IPH function defined on Then the following inequality holds for all :
Proposition 2.2 can be easily shown by using the Proposition 2.1 (see ).
Let be an IPH function, and let be integrable on Then
for all and this inequality is sharp.
Unlike the previous work, inequality (2.7) (obtained for IPH functions) and inequalities in the type of (2.7) (will be obtained for different function classes) are going to be inquired for more general the sets not for the set. will be certainly different for each function class.
Let be a closed domain, that is, and let be positive number. Let be the set of all points such that
In [6, Proposition 3.2], the proposition has been given for , the same proposition is defined for as follows, and its proof is similar.
Let be an IPH function defined on If the set is nonempty and is integrable on , then
We had proved a proposition in  by using a function and we get a right-hand side inequality, similar to (2.7).
Let be an IPH function, and let be integrable function on Then
For every the inequality
2.2. Increasing Positively Homogeneous Functions and Hermite-Hadamard Type Inequalities
A function is called increasing radiant (InR) function if
(1) is increasing;
(2) is radiant; that is, for all , and
Consider the coupling function defined on
Denote by the function defined on by the formula
It is known that the set
is supremal generator of all increasing radiant functions defined on (see)
Note that for we get .
The very important property for InR functions is given here in after. It can be easily proved.
Let be an InR function defined on Then the following inequality holds for all :
By using [9, Proposition 2.5], the following proposition is proved.
Let be InR functions and integrable on . Then
for all This inequality is sharp for any since one has the inequality in  for
We determine the set for InR functions. Let be the set of all points such that
which is given in [9, Proposition 3.1] can be generalized for .
Let be an InR function defined on If the set is nonempty and is integrable on then
The proof of the proposition can be made in a similar way to the proof in [9, Proposition 3.1].
Now, we will study to achieve right-hand side inequality for InR functions.
First, Let us prove the auxiliary proposition.
Let be an InR function on Then the following inequalities hold for all :
Since is InR function on then
for all From this
If we consider the definition of then
Let be an InR function and integrable on , and
holds and is sharp since we get equality for
It follows from Proposition 2.8.
Let be an InR function and integrable on If and for all then
holds and is sharp.
2.3. Increasing Coradiant Functions and Hermit-Hadamard Type Inequalities
A function defined on a cone is called coradiant if
It is easy to check that is coradiant if and only if
We will consider increasing coradiant (ICR) function defined on the cone
Consider the function defined on
Recall that the set
is supremal generator of the class ICR functions defined on (see ).
The Hermit-Hadamard type inequalities have been obtained for ICR functions by using the following proposition in .
Let be an ICR function defined on Then the following inequality holds for all :
Let be ICR function and integrable on Then the following inequality holds for all :
and it is sharp.
The set is defined for ICR function, namely, denotes the set of all points such that
Let be an ICR function on If the set is nonempty and is integrable on , then
Let us define a new function such that
where is max-type function. By including the new function , we can achieve right-hand side inequalities for ICR functions, too.
Let function be an ICR function and integrable on Then
and for every the inequality
2.4. Increasing Convex Along Rays Functions and Hermit-HadamardType Inequalities
The Hermite-Hadamard type inequalities are studied for ICAR functions in . But only the functions which are defined on are considered.
In this article, the functions which are defined on are considered, and general results are found.
Let be a conic set. A function is called convex-along-rays if its restriction to each ray starting from zero is a convex function of one variable. In other words, it means that the function
is convex for each
In this paper we consider increasing convex-along- rays (ICARs) functions defined on
It is known that a finite ICAR function is continuous on the and lower semicontinuous on in .
Let us give two theorems which had been proved in [10, Theorems 3.2 and 3.4].
Let be the class of all functions defined by
where is a min-type function and A function is -convex if and only if is lower semicontinuous and ICAR.
Let be ICAR function, and let be a point such that for some Then the sup differential
is not empty and
Now, we can define the following theorem which is important to achieve Hermit-Hadamard type inequalities for ICAR functions.
Let be a finite ICAR function defined on Then for each there exists a number such that
The result follows directly from Theorem 2.16
We will apply Theorem 2.17 in the study of Hermit-Hadamard type inequalities for ICAR functions.
Let , be ICAR function. Then the following inequality holds for all :
It follows from Theorem 2.17.
Formula (2.44) can be made simply with the sets .
Let be a bounded set such that and
Let the set be nonempty, and let be a continuous ICAR function defined on Then the following inequality holds:
Let It follows from (2.43) and the definition of that
Since is compact (see ) and is continuous (finite ICAR functions is continuous), it follows that the maximum in (2.46) is attained.
Inequalities (2.9), (2.18), (2.35), and (2.46), which are obtained for different convex classes, are actually different, even if they appear to be the same. The reason is that these are determined with the (2.8), (2.17), (2.34), and (2.45) formulas appropriate for the sets of and also yielding different sets.
The results of different classes of convex functions are defined for same triangle region
The inequalities (2.7) and (2.10) have been defined for IPH functions. The inequalities are examined for the region in . If we combine two results, then we get
If we study the set for IPH functions, a point belongs to if and only if
That is, the set is intersection with the set and the parabola by formula (3.3).
Let us consider the InR functions for same region The inequality (2.16) has been examined for and the following inequality has been obtained in :
Let us study on the right-hand side inequality (2.26), which is obtained in this article, for same region , which has been defined as follows:
We will separate two sets:
In this case, we get
Thus, the inequality (2.26) becomes
for all it is held.
The set can be defined for InR functions such that, a point belongs to if and only if
The inequalities (2.33) and (2.35) had been obtained for ICR functions. If these inequalities are examined for the same triangle region then the following inequality is obtained in :
The set has been obtained for ICR functions as formula (2.34). Formula (2.34) becomes formula (3.11) for the triangle region That is a point belongs to if and only if
Lastly, formula (2.44) has been defined for ICAR functions. Now, we will define the same formula for the triangle region :
and the inequality is held for all , where is parameter which depends on (see ).
ICAR functions had been studied for the set which is determined by formula (2.45). If in , then the set is special case of the given formula (2.8). Then a point belongs to if and only if
In other words, belongs to the parabola by formula (3.13).
Hermite-Hadamard type inequalities are investigated for specific functions classes. One of these functions classes is abstract convex functions. The deriving Hermite-Hadamard type inequalities for IPH, InR, ICR, and ICAR functions, which are important classes of abstract convex functions, are investigated by different authors [6–10].
In this article, this problem is considered entirely; findings from [6–10] are summarized; new results are found for some classes; results of some classes are generalized. For example, all results are found for more general case, not all for . Even though the results, (2.9), (2.18), (2.35), (2.46), are similar in appearance, they represent different inequalities, since the sets, which are defined with formulas (2.8), (2.17), (2.34), and (2.45), for different classes, are different.
Right-hand side inequalities, which are found for InR functions classes in , are considered here as well; more general results are found with the support of functions and explained as Proposition 2.9.
ICAR functions, which are studied in , are investigated on here, and results are explained in Proposition 2.18 . The inequality, which is explained in formula (2.44), is a new inequality for these functions classes.
Finally, all the results are explained for the same region given on . Formulas (3.2), (3.8), (3.10), and (3.12) are concrete results of Hermit-Hadamard type inequalities of different abstract convex function classes on given triangle region. Formulas (3.3), (3.9), (3.11), and (3.13) are concrete explanations of sets in this region.
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The first author was supported by the Scientific Research Project Administration Unit of Mersin University (Turkey). The second author was supported by the Scientific Research Project Administration Unit of Akdeniz University (Turkey).
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Adilov, G.R., Kemali, S. Abstract Convexity and Hermite-Hadamard Type Inequalities. J Inequal Appl 2009, 943534 (2009). https://doi.org/10.1155/2009/943534
- Convex Function
- Function Class
- Type Inequality
- Homogeneous Function
- Coupling Function