Certain Aspects of Univalent Function with Negative Coefficients Defined by Bessel Function

ABSTRACT

Key words: In recent years, applications of Bessel functions have been effectively used in the modelling of chemical engineering processes and theory of univalent functions.In this paper, we study a new class of analytic and univalent functions with negative coefficients in the open unit disk defined by Modified Hadamard product with Bessel function. We obtain coefficient bounds and exterior points for this new class.

Keywords:
Analytic function; Univalent function; Extreme point; Bessel function; Subordination; Hadamard Product

INTRODUCTION

Many important functions in applied sciences are defined via improper integrals or series (or infïnite products). The general name of these important functions are called special functions. Bessel functions are important special functions which are playing the important role in studying solutions of differential equations. Especially, the linear PDE describing various chemical transfer processes, allow the exact solution expressed in terms of one special kind of Bessel's functions and they are associated with a wide range of problems in important areas of mathematical physics, modelling of transfer processes in chemical engineering as well as in the related fields like hydrodynamics, heat transfer, diffusion, bioprocesses and so on. By using the method of seperation of variables, exact solution in terms of Bessel function can be used to calculate several important parameters which are needed in design and construction of chemical engineering apparatuses and equipment like heat exchangers and their components. Typical example for the efficiency calculation is applied in Brazilian powdered milk plant [¹¹]. In another case when the Bessel functions arises is heat transfer modelling which considered in [⁶]. Here the problem of cross-flow streaming of heated object with large value of length to diameter ratio (like thermoanemometer) is solved for small Pe numbers using the theory of analytic functions.

Recently Deniz [⁹] has studied the following: The generalized Bessel function of the first kind of order , is defined as function , has the familiar representation as follows

(1.1)

Where stands for the Euler gamma function. The series (1.1) permits the study of Bessel function in a unified manner. It is called the particular solution of the following second-order linear homogeneous differential equation (see for details[⁵]):

(1.2)

. Also in Deniz et al. [⁷] and Deniz [⁸] (see also[¹, ³, ⁴, ¹⁰] studied the function defïned, in terms of the generalized Bessel function by the transformation

(1.3)

By using the well known Pochhammer symbol (or the shifted factorial) is defined in terms of the Euler function, by

it being understood conventionally that . We obtain the following series representation for the function given by (1.3):

(1.4)

where , and . For convenience, we write . Next, we introduce a operator , which is defined by the Hadamard product

(1.5)

where

It is easy to verify that from (1.5)

where . In fact the function given by (1.5) is an elementary transformation of the generalized hypergeometric function. Hence, it is easy to see that and also . Let be the class of all analytic functions

(1.6)

in the unit disk . Let be the subclass of consisting of univalent functions. Suppose for , and denote the subclasses of consisting of functions which satisfy the following inequalities:

are, respectively, starlike and convex of order in In particular, we set and . Let denote the subclasses of consisting of functions given by

(1.7)

with negative coefficients. Silverman[¹³] introduced and investigated the following subclasses of the function class :

(1.8)

For given by (1.6) and is given by , the Hadamard product (or convolution) of and is given by

Recently Shanmugam et al.[¹²] and [¹⁵] have studied the following:

Definition 1.1 Let, , , and, a function is said to be in the classif it satisfies the following inequality:

(1.9)

where .

Lemma 1.2 [²] Let . Then if and only if

Lemma 1.3 [²] Let andare real number. Thenif and only if.

2 Coefficient Bounds and Extreme Points

We obtain the necessary and sufficient condition and extreme points for the function in the class .

Theorem 2.1 Let , , , and.The function defined by equation (1.6) is in the classif and only if

(2.1)

Proof. From the definition, we have

From Lemma 1.3, we have

, or equivalently

(2.2)

Let

and By Lemma 1.2, equation (2.2) is equivalent to

, for .But

or

which is equivalent to

conversely, suppose that the equation (2.1) holds good, then we have to prove that

Now choosing the values of z on the positive real axis where , the above inequality reduces to

Since , the above inequality reduces to

Letting , we get the desired result.

Corollary 2.2 Let , , , and, if , then

Theorem 2.3 Let , , , , , and.If and

Then if and only if it can be expressed in the form

Proof. Let , where and or . Then

But

Using Theorem 2.1, we have .

Conversely, Let us assume that is of the form (1.6) belongs to . Then

Setting

we have

Hence the proof.

3 Growth and Distortion Theorem

Theorem 3.1 If andthen

(3.1)

Equality in (3.1) holds true for the function given by

(3.2)

Proof. we only prove the second part of the inequality in (3.1), since the first part can be derived by using similar arguments. If , by using Theorem 2.1, we find that

which readily yields the following inequality

(3.3)

Moreover it follows that

which proves the second part of the inequality in (3.1).

Theorem 3.2 If andthen

(3.4)

Equality in (3.1) holds true for the function given by (3.2)

Proof. Our proof of Theorem 3.2 is much akin to that of Theorem 3.1. Indeed, since , it is easily verified from (1.7) that

(3.5)

and

(3.6)

The assertion (3.4) of Theorem 3.2 would now follow from (3.5) and (3.6) by means of a rather simple consequence of (3.3) given by

(3.7)

The completes the proof of Theorem 3.2.

4 Hadamard Product

Theorem 4.1 Let

belongs to . Then the Hadamard Product of and given by

Proof. Since and belongs to we have

and

and by applying the Cauchy-Schwarz inequality, we have

However,we obtain

Now we have to prove that

Hence the proof.

5 Application of the Fractional Calculus

Various operators of fractional calculus (i.e fractional derivative and fractional integral) have been rather extensively studied by many researchers(see for example [¹⁴]). Each of these theorems would involve certain operator of fractional calculus which are defined as follows.

Definition 5.1 The fractional integral operator of orderis defined for a functionby

(5.1)

where is analytic function in a simply connected region of z-plane containing the origin and the multiplicity of is removed by requiring to be read when .

Definition 5.2 The fractional derivative of order is defined for a functionby

(5.2)

where is analytic function in a simply connected region of z-plane containing the origin and the multiplicity of is removed by requiring to be read when .

Definition 5.3 The fractional derivative of order is defined by

(5.3)

From definition (5.1) and (5.2), after a simple computation we obtain

(5.4)

(5.5)

Now using equations (5.4) and (5.5), Let us prove the following theorems:

Theorem 5.4 LetThen

(5.6)

(5.7)

The inequalities (5.6) and (5.7) are attained for the function f given by

(5.8)

Proof. From Theorem 2.1, we obtain

(5.9)

Using equation (5.5), we obtain

(5.10)

such that

where is a decreasing function of n and .

Using equation (5.9) and (5.10), we obtain

Which is the equation (5.6). Similarly we can get equation(5.7).

Theorem 5.5 Let Then

(5.11)

(5.12)

The inequalities (5.11) and (5.12) are attained for the function f given by

(5.13)

Proof. Using equation (5.5), we obtain

(5.14)

such that

where is a decreasing function of n and .

Using equation (5.9) and (5.114), we obtain

Which is the equation (5.11). Similarly we can get equation (5.12).

Corollary 5.6 For every , we have

and

Proof. By Definition 5.1 and Theorem 5.4 for , we have , the result is true. Also by Definition 5.2 and Theorem 5.5 for , we have

Hence the result is true.

6 Radii of close-to-convexity Starlikeness and Convexity Theorem

Theorem 6.1 Let the function defined bybe in the classThenis close to convex of orderinwhere

The result is sharp for the function given by

Proof. It is sufficient to show that , for ).

Hence

and if

(6.1)

Thus by Theorem 2.1, (6.1) will holds true if

(or) if

The theorem follows easily from previous equation.

Theorem 6.2 Let the function defined bybe in the classThenis starlike of orderinwhere

The result is sharp for the function given by

Proof. If is starlike it is sufficient to show that , for ). Since

Since

if

(6.2)

Thus by Theorem 2.1, (6.2) will holds true if

(or) if

The theorem follows easily from previous equation.

Theorem 6.3 Let the function defined bybe in the classThenis convex of orderinwhere

The result is sharp for the function given by

Proof. Since is convex it is enough to show that , for . Since

Thus

if

(6.3)

Hence by Theorem 2.1, (6.3) will holds true if

(or) if

The theorem follows easily from previous equation.

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Publication Dates

• Publication in this collection
  2016

History

• Received
  03 Feb 2016
• Accepted
  14 July 2016