Keywords: generalized convex functions, trigonometrically ρ convex functions, supporting functions, integral inequalities
American Journal of Mathematical Analysis, 2014 2 (1),
pp 47.
DOI: 10.12691/ajma212
Received January 19, 2014; Revised February 13, 2014; Accepted February 17, 2014
Copyright © 2014 Science and Education Publishing. All Rights Reserved.
1. Introduction and Preliminaries
Trigonometrically  convex functions play a central role in many various aspects in the theory of entire functions ( of order ) and in the theory of cavitational diagrams for hydroprofiles, see for example ^{[4, 5, 6]}. In what follows, we shall be concerned with real finite functions defined on a finite or infinite interval In this section we present the basic definitions and results which will be used later, for more information see ^{[5, 6]}.
Definition 1.1. A function is said to be trigonometrically  convex, if for any arbitrary closed subinterval of I such that the graph of for lies nowhere above the  trigonometric function, determined by the equation:
where A and B are chosen such that
Equivalently, if for all
Definition 1.2. A function
is said to be supporting function for at the point if
Theorem 1.1. ^{[1]} A function is trigonometrically  convex on if and only if there exists a supporting function for at each point
Property 1.1. ^{[1]} If is differentiable trigonometrically  convex function, then the supporting function for at the point has the form:
To keep the paper self contained we present the following results ^{[7]}.
Theorem 1.2. If is differentiable function. Then is convex if and only if is increasing.
Theorem 1.3. A function is convex if and only if there is an increasing function and a point such that for all
Theorem 1.4. If and are both nonnegative, increasing (decreasing), and convex, then also preserves these three properties.
In ^{[3]}, B. J. Andersson established the following theorem
Theorem 1.5. Let be convex functions, defined in and for which
If then
 (1) 
2. Results
In this section we derive a similar result to Andersson's inequality for trigonometrically  convex functions. For more inequalities, one may refer to ^{[2]}.
Theorem 2.1. Let be continuous convex functions, defined in for which
and let be a nonnegative,  periodic, differentiable, and trigonometrically  convex function defined on such that: and
then
Proof. As is trigonometrically  convex function, then from Definition 1.2, it follows that:
Since is differentiable and then from Property 1.1, the supporting function for at the point can be written in the form
Consequently,
 (2) 
As by using (2), one has:
 (3) 
Using the following substitution
 (4) 
and let
 (5) 
then it follows that:
and is an increasing function in Using Theorem 1.2, one obtains that is a convex function in
Let
we obseve that the functions satisfy all assumptions of the Theorem 1.5 in the interval
Hence, using (1), one obtains:
 (6) 
Now using (4), (5), and (6), then (3) turns out to:
where
Hence, the claim.
Now, we are in a position to prove the following main theorem.
Theorem 2.2. Let be convex functions, defined in for which
and let be a nonnegative,  periodic, differentiable, and trigonometrically  convex function defined on such that:
Then, one has the following sharp inequality:
 (7) 
Proof. The proof will be divided into 4 steps.
Let M denote the class of convex functions of the theorem.
Step 1. If then is increasing.
Since is convex in and then from Theorem 1.3 there is an increasing function
 (8) 
Now, suppose that for some
As is increasing, then for all therefore,
From (8) it follows, which contradicts the fact that
Thus,
Now, let if then using (8), one has:
Hence, is an increasing function.
For the next steps, let
 (9) 
and
 (10) 
Step 2. M is closed under multiplication.
From (9), it follows that:
is nonnegative, increasing, and convex function satisfies
Thus,
Hence, from Theorem 1.4, M is closed under multiplication.
This confirm that this property is independent of the choice of
 (11) 
 (12) 
From the convexity of it follows that the graph of must intersect the straight line of in a unique point q as shown in Figure 1.
If x lies in [0,p], then obviously from (10) it followos that
Otherwise, if one has:
Using (11), one obtains:
Thus,
Hence
As is differentiable trigonometrically  convex function and according to Property 1.1, for convenience, we denote
 (13) 
as the supporting function for at the point
But, from Definition 1.2, one obtains:
 (14) 
hence we can go to:
Step 4. We show that If then
 (15) 
From Step 2, we observe:
Using (14), one has,
 (16) 
Let
 (17) 
Using (10) and (12), it follows that
Since and from Step 3, we infer that
Thus
 (18) 
Hence, from (16), (17) and (18), we obtain the required inequality (15).
Now, we prove the main inequality (7).
From Step 2, we have
Thus, using (15), one has
Again,
Hence, from (18) it follows that
Repeating the above argument and using (18) each time, then from (9) and (13) one obtains:
Using the following substitution
we get
and the theorem is proved.
Note: the equality in (7) occurs if
Acknowledgement
The author wishes to thank the anonymous referees for their fruitful comments and suggestions which improved the original manuscript.
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