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The aim of this paper is to prove that the average function of a trigonometrically ρ-convex function is trigonometrically ρ-convex. Furthermore, we show the existence of support curves implies the trigonometric ρ-convexity, and prove an extremum property of this function.

In 1908, Phragmén and Lindelöf ( See, e.g. [

has the following property:

If, and is the function of the form

(such functions are called sinusoidal or ρ-trigonometric) which coincides with at and at, then for we have

This property is called a trigonometric ρ-convexity ([1,2]).

In this article we shall be concerned with real finite functions defined on a finite or infinite interval

A well known theorem [

In Theorem 3.1, we prove this result in case of trigonometrically ρ-convex functions. In Theorem 3.2, we prove the extremum property [

In this section we present the basic definitions and results which will be used later , see for example ([1,2], and [6-9]).

Definition 2.1. A function is said to be trigonometrically ρ-convex if for any arbitrary closed subinterval of such that , the graph of for lies nowhere above the ρ-trigonometric function, determined by the equation

where and are chosen such that and

Equivalently, if for all

The trigonometrically ρ-convex functions possess a number of properties analogous to those of convex functions.

For example: If is trigonometrically ρ-convex function, then for any such that the inequality holds outside the interval

Definition 2.2. A function

is said to be supporting function for at the point if

That is, if and agree at and the graph of does not lie under the support curve.

Remark 2.1. If is differentiable trigonometrically ρ-convex 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 such that and as

Then taking the limit of both sides as and from (1), one obtains

Thus, the claim follows.

Theorem 2.1. A trigonometrically ρ-convex function has finite right and left derivatives at every point and for all

Theorem 2.2. Let be a two times continuously differentiable function. Then is trigonome-trically ρ-convex on if and only if for all

Property 2.1. Under the assumptions of Theorem 2.1, the function is continuously differentiable on with the exception of an at most countable set.

Property 2.2. A necessary and sufficient condition for the function to be a trigonometrically ρ-convex in is that the function

is non-decreasing in.

Lemma 2.1. Let be a continuous, - periodic function, and the derivative exists and piecewise continuous function and let be a set of discontinuity points for If

and where

Then is trigonometrically ρ-convex on.

Proof. Consider

Two cases arise, as follows.

Case 1. Suppose Using (5), we observe

From (3), we get

So, the function is non-decreasing in Case 2. Let and

Differentiating both sides of (5) with respect to one has

Using (4), one obtains

Thus, is non-decreasing function in

Therefore, from Property 2.2, we conclude that the function is trigonometrically ρ-convex on.

Theorem 3.1. A function is trigonometrically ρ-convex on if and only if there exists a supporting function for at each point in.

Proof. The necessity is an immediate consequence of F. F. Bonsall [

To prove the sufficiency, let be an arbitrary point in and has a supporting function at this point. For convenience, we shall write the supporting function in the follwoing form:

where is a fixed real number depends on and.

From Definition 2.2, one has

It follows that,

For all choose any such that and with and let

Applying (6) twice at and at yields

Multiplying the first inequality by the second by and adding them, we obtain

Consequently

for all which proves that the function is trigonometrically ρ-convex on.

Hence, the theorem follows.

Remark 3.1. For a trigonometrically ρ-convex function, the constant in the above theorem is equal to if is differentiable at the point

in, otherwise,

Theorem 3.2. Let be a trigonometrically ρ-convex function such that and let be a supporting function for at the point Then the function

has a minimum value at

Proof. From Definition 2.2, we have

and

and can be written in the form

where and

Using (9), one obtains

Consequently,

Using (7) at the function becomes

But from (8) ,we observe for all.

Now using (10) and (11), it follows that

for all.

Hence, the minimum value of the function

occurs at.

Theorem 3.3. Let be a non-negative, 2π- periodic, and trigonometrically ρ-convex function with a continuous second derivative on and let be a 2π-periodic function defined in as follows

If and

Then, is trigonometrically ρ-convex function.

Proof. The proof mainly depends on Lemma 2.1. So, we show that the function satisfies all conditions in this lemma.

Suppose that

It is obvious that,

First, we study the behavior of the function inside the interval.

It is clear from (12) that s is an absolutely continuous function, has a derivative of third order.

But from the periodicity of and (13), we get

Using the following substitution.

It follows that, can be written as

and.

Consequently,

Since is non-negative, trigonometrically ρ-convex function, and then from Theorem 2.2 and (16) it follows that

Second, we prove that

From the definition of in (14) and the periodicity of we observe that and.

Again using (14), we have

Thus, from (15) and (19), one has, and

.

Hence, from (13), we infer that

and the inequality in (18) is proved.

Now using (17), (18), and Lemma 2.1, we conclude that is trigonometrically ρ-convex function, and the theorem is proved.

The author wishes to thank the anonymous referees for their fruitful comments and suggestions which improved the original manuscript.