Abstract Convexity and Hermite-Hadamard Type Inequalities
© G. R. Adilov and S. Kemali. 2009
Received: 24 February 2009
Accepted: 8 May 2009
Published: 14 May 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]).
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 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 .
for all and
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.
Proposition 2.2 can be easily shown by using the Proposition 2.1 (see ).
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.
In [6, Proposition 3.2], the proposition has been given for , the same proposition is defined for as follows, and its proof is similar.
We had proved a proposition in  by using a function and we get a right-hand side inequality, similar to (2.7).
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
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.
By using [9, Proposition 2.5], the following proposition is proved.
for all This inequality is sharp for any since one has the inequality in  for
which is given in [9, Proposition 3.1] can be generalized for .
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.
holds and is sharp since we get equality for
It follows from Proposition 2.8.
holds and is sharp.
2.3. Increasing Coradiant Functions and Hermit-Hadamard Type Inequalities
We will consider increasing coradiant (ICR) function defined on the cone
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 .
and it is sharp.
where is max-type function. By including the new function , we can achieve right-hand side inequalities for ICR functions, too.
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.
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].
where is a min-type function and A function is -convex if and only if is lower semicontinuous and ICAR.
Now, we can define the following theorem which is important to achieve Hermit-Hadamard type inequalities for ICAR functions.
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.
It follows from Theorem 2.17.
Formula (2.44) can be made simply with the sets .
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.
That is, the set is intersection with the set and the parabola by formula (3.3).
In this case, we get
for all it is held.
and the inequality is held for all , where is parameter which depends on (see ).
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.
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).
- Dragomir SS, Pearce CEM: Quasi-convex functions and Hadamard's inequality. Bulletin of the Australian Mathematical Society 1998,57(3):377–385. 10.1017/S0004972700031786MathSciNetView ArticleMATHGoogle Scholar
- Dragomir SS, Pečarić J, Persson LE: Some inequalities of Hadamard type. Soochow Journal of Mathematics 1995,21(3):335–341.MathSciNetMATHGoogle Scholar
- Gill PM, Pearce CEM, Pečarić J: Hadamard's inequality for -convex functions. Journal of Mathematical Analysis and Applications 1997,215(2):461–470. 10.1006/jmaa.1997.5645MathSciNetView ArticleMATHGoogle Scholar
- Pearce CEM, Rubinov AM: -functions, quasi-convex functions, and Hadamard-type inequalities. Journal of Mathematical Analysis and Applications 1999,240(1):92–104. 10.1006/jmaa.1999.6593MathSciNetView ArticleMATHGoogle Scholar
- Rubinov AM, Dutta J: Hadamard type inequality for quasiconvex functions in higher dimensions. Journal of Analysis and Applications 2002, 270: 80–91. 10.1016/S0022-247X(02)00050-1MathSciNetView ArticleMATHGoogle Scholar
- Adilov GR, Kemali S: Hermite-Hadamard-type inequalities for increasing positively homogeneous functions. Journal of Inequalities and Applications 2007, 2007:-10.Google Scholar
- Adilov GR: Increasing co-radiant functions and Hermite-Hadamard type inequalities. to apprear in Mathematical Inequalities and Applications to apprear in Mathematical Inequalities and Applications
- Dragomir SS, Dutta J, Rubinov AM: Hermite-Hadamard-type inequalities for increasing convex-along-rays functions. Analysis 2004,24(2):171–181.MathSciNetMATHGoogle Scholar
- Sharikov EV: Hermite-Hadamard type inequalities for increasing radiant functions. Journal of Inequalities in Pure and Applied Mathematics 2003.,4(2, article 47):Google Scholar
- Rubinov A: Abstract Convexity and Global Optimization, Nonconvex Optimization and Its Applications. Volume 44. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:xviii+490.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.