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

Figure 1.

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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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[4]	Avhadief F. G. and Maklakov D. V., “A Theory of Pressure Envelopes for Hydrofoils,” Journal of Ship Resrarch., 42, 81-102, 1995.
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[5]	Levin B. Ya., “Lectures on Entire Functions,” American Mathematical Society, 1996.
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