On Certain Properties for Two Classes of Generalized Convex Functions

Abstract

Two classes of generalized convex functions in the sense of Beckenbach are considered. For both classes, we show that the existence of support curves implies their generalized convexity and obtain an extremum property of these functions. Furthermore, we establish Hadamard’s inequality for them.

1. 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 A real valued function of a single real variable defined on is said to be convex if for all and one has the inequality

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 and of the plane there exists exactly one member of the family with a graph joining these points.

More precisely, let be a family of continuous functions defined in a real interval . A function is said to be sub -function if, for any with , there is a unique member of satisfying(i) and ,(ii) for all .

The sub -functions possess various properties analogous to those of classical convex functions [2–7]. For example, if is sub -function, then, for any , the inequality holds outside the interval

Theorem 1. A sub -function has finite left and right derivatives at every point , and for all .

Property 1. Under the assumptions of Theorem 1, the function is continuously differentiable on with the exception of an at-most countable set.

Of course mathematicians were able before 1937 to generalize the notion of convex functions [8–10]. Full details could be found in two classic books [11, 12] or in the new monographs like [13].

In this paper, we deal just with generalized convexity in the sense of Beckenbach. For particular choices of the two-parameter family , one considers two classes of generalized convex functions:(i), where is a fixed constant.(ii).

The following double inequalityis well known in the literature as Hadamard’s inequality or, as it is quoted for historical reasons [14], the Hermite-Hadamard inequality, where 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, [15–18].

A basic theorem [11] in the theory of convex functions states that a necessary and sufficient condition in order that the function 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.

2. 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 is said to be sub -function on , if for any arbitrary closed subinterval of the graph of for lies nowhere above the functionwhere and are chosen such that and .
Equivalently, for all

Note that the condition for all in is necessary and sufficient in order that the twice differentiable function be sub -function on .

Definition 3. Let be a sub -function.
A functionis said to be a supporting function for at the point , ifThat is, if and agree at , the graph of does not lie under the support curve.

Proposition 4. If 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 and Then, taking the limit of both sides as and from (5), one obtainsThus, the claim follows.

Definition 5. A positive function is called sub -function on , if for any with the graph of for lies on or under the functionwhere and are taken so that , and
Equivalently, for all

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 be a two-time continuously differentiable function. Then, is sub -function on if and only if for all .

Definition 6. Let be a sub -function.
A functionis said to be a supporting function for at the point , ifThat is, if and agree at , the graph of lies on or above the support curve.

Proposition 7. If 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 and Then, taking the limit of both sides as and from (14), one obtainsThus, the claim follows.

In the literature, the logarithmic mean of the positive real numbers is defined asThe logarithmic mean proves useful in engineering problems involving heat and mass transfer.

3. Results

Theorem 8. A function 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 hasIt follows thatFor all with and with letApplying (25) twice at and at yieldsMultiplying the first inequality by and the second by and adding them, we obtainConsequently,which proves that the function is sub -function on .
Hence, the theorem follows.

Remark 9. For a sub -function , the constant in the foregoing theorem is equal to if is differentiable at the point ; otherwise, .

Theorem 10. Let be a sub -function and with , and let be a supporting function for at the point Then, the functionhas a minimum value at .

Proof. From Definition 3, we haveand can be written in the formUsing (33), one obtainsConsequently,Using (31) at , the function becomesBut from (32), we observeNow using (35) and (36), it follows thatHence, the minimum value of the function occurs at .

Theorem 11. Suppose is sub -function, and with .
Then, one has the inequality

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 and nowhere below any supporting function: at the point .
Hence,and thus Using (40), one hasUsing (41) and (34), one obtainsBut from (32), we observe thatTaking the maximum of the term in (43) and (45) for and from (46), it follows thatHence, from (43), (44), and (47), we get the desired inequality (39).

Theorem 12. A function is a 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 6, one hasIt follows thatAs is a positive function, we infer thatFor all with and with letApplying (51) twice at and at yieldsMultiplying the first inequality by and the second by and adding them, we obtainConsequently,which proves that the function is a sub -function on .
Hence, the theorem follows.

Remark 13. For a sub -function , the constant in the preceding theorem is equal to if is differentiable at the point ; otherwise,

Theorem 14. Let be a sub -function, and with and let be a supporting function for at the point . Then, the function has a minimum value at .

Proof. From Definition 6, we haveand can be written in the formUsing (59), one obtainsConsequently,One can easily check that does not depend on . That is, depends only on and .
Using (57) at , the function becomesBut from (58), we observeNow using (61) and (62), it follows thatHence, the minimum value of the function occurs at .

Theorem 15. Assume is a sub -function, and with .
Then, one has the inequality

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 and nowhere below any supporting function: at the point .
Thus,Using (66), one hasUsing (67) and (60), one obtainsIt is easy to notice thatBut from (58), we observe thatTaking the maximum of the term in (68) and (71) for and from (72), it follows thatHence, from (68), (69), and (73), we get the required inequality (65).

Remark 16. Recall that a positive function is said to be log-convex or multiplicatively convex if is convex, equivalently, if for all and one has the inequalityusing the following substitution:Then, from (15), the function becomesThus, from Definition 5, it follows that a function is a sub -function on , if for all and one has the inequalityTherefore, from (74) and (77), we conclude that the sub -functions are exactly the “standard” log-convex functions. On the other hand, this class of sub -functions can be considered as another point of view for log-convex functions.

4. 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.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.