On Certain Properties of Trigonometrically ρ-Convex Functions

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.


Introduction
In 1908, Phragmén and Lindelöf ( See, e.g.[1]) showed that if   F z 0 < < is an entire function of order   , then its indicator which is defined as: 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 [3] in the theory of ordinary 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 : ( f a f at each point x in   , .a b In Theorem 3.1, we prove this result in case of trigonometrically ρ-convex functions.In Theorem 3.2, we prove the extremum property [4] of convex functions in case of trigonometrically ρ-convex functions.And finally in Theorem 3.3, we show that the average function [5] of a trigonometrically ρ-convex function is also trigonometrically ρ-convex.

Definitions and Preliminary Results
In this section we present the basic definitions and results which will be used later , see for example ( [1,2], and [6][7][8][9]).
Definition 2.1.A function is said to be trigonometrically ρ-convex if for any arbitrary closed subinterval , the graph of f x for x , u v  lies nowhere above the ρ-trigonometric function, determined by the equation where A and are chosen such that The trigonometrically ρ-convex functions possess a number of properties analogous to those of convex functions.


For example: That is, if Proof.The supporting function for u a b  at the point can be described as follows: where such that 0 < and as Then taking the limit of both sides as v and from (1), one obtains , Thus, the claim follows.Theorem 2.1.A trigonometrically ρ-convex function has finite right and left derivatives with the exception of an at most countable set.
Property 2.2.A necessary and sufficient condition for the function f x to be a trigonometrically ρ-convex is non-decreasing in .Lemma 2.1.Let be a continuous, periodic function, and the derivative f where and Two cases arise, as follows.Case 1. Suppose x x M  Using (5), we observe exists and piecewise continuous function and let M be a set of discontinuity points for , Differentiating both sides of (5) with respect to x one has

 
x Thus,  is non-decreasing function in Therefore, from Property 2.2, we conclude that the function Proof.The necessity is an immediate consequence of F. F. Bonsall [10].

Main Results
To prove the sufficiency, let x be an arbitrary point in   , a b and f has a supporting function at this point.For convenience, we shall write the supporting function in the follwoing form: x and where x f K is a fixed real number depends on f .From Definition 2.2, one has = , and , .
Multiplying the first inequality by the second by and adding them, we obtain   , a b Hence, the theorem follows.Remark 3.1.For a trigonometrically ρ-convex function , the constant Let be a trigonometrically ρ-convex function such that and let be a supporting function for and and can be written in the form where , , Using ( 7) at , 2 . Now using (10) and (11), it follows that  But from the periodicity of F x and (13), we get From the definition of   and the inequality in (18) is proved.Now using (17), (18), and Lemma 2.1, we conclude that x is trigonometrically ρ-convex function, and the theorem is proved.
are called sinusoidal or ρ-trigonometric) which coincides with  at  and at  , then for 

Theorem 3 . 1 .
A function is trigonometrically ρ-convex on   , a b   if and only if there exists a supporting function for x in f x at each point   , a b .

Theorem 3 . 3 .
Let f x   be a non-negative, 2πperiodic, and trigonometrically ρ-convex function with a continuous second derivative on and let function.