Keywords: generalized convex functions, trigonometrically ρ- convex functions, supporting functions, integral inequalities
American Journal of Mathematical Analysis, 2014 2 (1),
pp 4-7.
DOI: 10.12691/ajma-2-1-2
Received January 19, 2014; Revised February 13, 2014; Accepted February 17, 2014
Copyright © 2013 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 non-negative, 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 non-negative,
- 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 non-negative,
- 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 non-negative, 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.
References
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