On Certain Properties for Two Classes of Generalized Convex Functions

and Applied Analysis 3 Definition 6. Let f : I → (0, ∞) be a sub E-function. A function T u (x) = Ae Bx (16) is said to be a supporting function for f(x) at the point u ∈ I, if


Introduction
The convexity of functions plays a central role in many various fields, such as in economics, mechanics, biological system, optimization, and other areas of applied mathematics.Throughout this paper, let  be a nonempty, connected, and bounded subset of R. A real valued function () of a single real variable  defined on  is said to be convex if for all , V ∈  and  ∈ [0, 1] one has the inequality  ( + (1 − ) V) ≤  () + (1 − )  (V) . ( At the beginning of the 20th century, many generalizations of convexity were extensively introduced and investigated in a number of ways by numerous authors in the past and present.One way to generalize the definition of a convex function is to relax the convexity condition (1) (for a comprehensive review, see the monographs [1]).
As it is well known, the notion of the ordinary convexity can be expressed in terms of linear functions.An important direction for generalization of the classical convexity was to replace linear functions by another family of functions.For instance, Beckenbach and Bing [2,3] generalized this situation by replacing the linear functions with a family of continuous functions such that for each pair of points  1 ( 1 ,  1 ) and  2 ( 2 ,  2 ) of the plane there exists exactly one member of the family with a graph joining these points.
In this paper, we deal just with generalized convexity in the sense of Beckenbach.For particular choices of the 2 Abstract and Applied Analysis two-parameter family {()}, one considers two classes of generalized convex functions: The following double inequality is well known in the literature as Hadamard's inequality or, as it is quoted for historical reasons [14], the Hermite-Hadamard inequality, where  :  → R is a convex function and ,  ∈  with  < .This inequality has evoked the interest of many mathematicians; for new generalizations, extensions, and numerous applications, see, for example, [15][16][17][18].A basic theorem [11] in the theory of convex functions states that a necessary and sufficient condition in order that the function  :  → R be convex is that there is at least one line of support for  at each point  in .
In this paper, we prove analogues of this result for the classes of sub -functions and -functions.We also extend the extremum property (as stated in [19]) and the Hermite-Hadamard inequality.

Definitions and Preliminary Results
Inspired by these investigations, let us now introduce the basic definitions and results for the preceding two classes, respectively, of generalized convex functions in the sense of Beckenbach as will be used later in this note.Definition 2. A function  :  → R is said to be sub function on , if for any arbitrary closed subinterval [, V] of  the graph of () for  ∈ [, V] lies nowhere above the function where  and  are chosen such that () = () and (V) = (V).
Equivalently, for all  ∈ [, V] Note that the condition   ()− 2 () ≥ 0 for all  in  is necessary and sufficient in order that the twice differentiable function  :  → R be sub -function on .Definition 3. Let  :  → R be a sub -function.
A function is said to be a supporting function for () at the point  ∈ , if   () =  () , That is, if () and   () agree at  = , the graph of () does not lie under the support curve.
Proposition 4. If  :  → R is a differentiable sub -function, then the supporting function for () at the point  ∈  has the form Proof.The supporting function   () for () at the point  ∈  can be described as follows: where V ∈  and Then, taking the limit of both sides as V →  and from (5), one obtains Thus, the claim follows.
Definition 5. A positive function  :  → (0, ∞) is called sub -function on , if for any , V ∈  with  < V the graph of () for  ≤  ≤ V lies on or under the function where  and  are taken so that () = (), and (V) = (V).

Note the following:
(1) There is more than one formula for the function () other than that stated in (13); for example, or in a multiplicative form (2) Let  :  → (0, ∞) be a two-time continuously differentiable function.
Proof.The supporting function   () for () at the point  ∈  can be described as follows: where V ∈  and Then, taking the limit of both sides as V →  and from (14), one obtains Thus, the claim follows.
In the literature, the logarithmic mean of the positive real numbers ,  is defined as The logarithmic mean proves useful in engineering problems involving heat and mass transfer.

Results
Theorem 8.A function  :  → R is sub -function on  if and only if there exists a supporting function for () at each point  in .
Proof.The necessity is an immediate consequence of Bonsall [20].
To prove the sufficiency, let  be an arbitrary point in  and  has a supporting function at this point.For convenience, we will write the supporting function in the following form: where  , is a fixed real number depending on  and .
From Definition 3, one has It follows that For all , V ∈  with  < V and ,  ≥ 0 with  +  = 1 let Applying (25) twice at  =  and at  = V yields Multiplying the first inequality by sinh (V − ) and the second by sinh (V − ) and adding them, we obtain Consequently, Proof.The necessity is an immediate consequence of Bonsall [20].
To prove the sufficiency, let  be an arbitrary point in  and  has a supporting function at this point.For convenience, we will write the supporting function in the following form: where  , is a fixed real number depending on  and .
From Definition 6, one has It follows that As () is a positive function, we infer that For all , V ∈  with  < V and ,  ≥ 0 with  +  = 1 let Applying (51) twice at  =  and at  = V yields Multiplying the first inequality by (V − ) and the second by (V − ) and adding them, we obtain Consequently, which proves that the function () is a sub -function on .Hence, the theorem follows.
Remark 16.Recall that a positive function  :  → (0, ∞) is said to be log-convex or multiplicatively convex if log () is convex, equivalently, if for all , V ∈  and  ∈ [0, 1] one has the inequality

Conclusion
Various generalizations of convex functions have appeared in the literature.In this paper, two classes of generalized convex functions in the sense of Beckenbach are considered.Some properties and inequalities for these classes are established.
That is, if () and   () agree at  = , the graph of () lies on or above the support curve.
which proves that the function () is sub -function on .Hence, the theorem follows.Remark 9.For a sub -function  :  → R, the constant  , in the foregoing theorem is equal to   () if  is differentiable at the point  ∈ ; otherwise,  , ∈ [  − (),   Proof.Let  be an arbitrary point in (, ).As () is a sub -function, then from Definitions 2 and 3 we observe that the graph of () lies nowhere above the function Theorem 12.A function  :  → (0, ∞) is a sub -function on  if and only if there exists a supporting function for () at each point  in .
Remark 13.For a sub -function  :  → (0, ∞), the constant  , in the preceding theorem is equal to   () if  is differentiable at the point  ∈ ; otherwise,   − () ≤  , ≤   + ().Proof.Let  be an arbitrary point in (, ).As () is a sub function, then from Definitions 5 and 6 we observe that the graph of () lies nowhere above the function