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An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions

G. Murugusundaramoorthy*, T. Janani

School of Advanced Sciences, VIT University Vellore - 632014, Tamilnadu, India

Abstract: The purpose of the present paper is to investigate some characterization for generalized Bessel functions of first kind to be in the new subclasses of β uniformly starlike and β uniformly convex functions of order α. Further we point out consequences of our main results.

Keywords: univalent, starlike, convex, uniformly starlike functions, uniformly convex functions, Bessel functions

Received December 02, 2014; Revised December 22, 2014; Accepted January 13, 2015

Citation: G. Murugusundaramoorthy, T. Janani, An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions. Turkish Journal of Analysis and Number Theory, 3 (1). 1-6. January 2015. doi:10.12691/tjant-3-1-1

1. Introduction

Let be the class of analytic functions of the form

[picture] | (1)

As usual, we denote by the subclass of consisting of functions which are normalized by and also univalent in Denote by the subclass of consisting of functions whose nonzero coefficients from second on, is given by

[picture] | (2)

For functions given by (1) and given by we define the Hadamard product (or convolution) of f and g by

[picture] | (3)

A function is said to be starlike of order , if and only if

This function class is denoted by . We also write where denotes the class of functions such that is starlike with respect to the origin. A function is said to be convex of order if and only if

This class is denoted by Further, , the well-known standard class of convex functions. It is an established fact that

Let and are the class of starlike and convex functions of order , introduced and studied by Silverman [21].

The class was introduced by Kanas and Wi´sniowska [12], where its geometric definition and connections with the conic domains were considered. The class was defined pure geometrically as a subclass of univalent functions, that map each circular arc contained in the unit disk with a center , onto a convex arc. The notion of uniformly convex function is a natural extension of the classical convexity. Observe that, if then the center is the origin and the class reduces to the class of convex univalent functions . Moreover for corresponds to the class of uniformly convex functions introduced by Goodman [10,11], and studied extensively by Rønning [19,20]. The class is related to the class by means of the well-known Alexander equivalence between the usual classes of convex and starlike functions. Further the analytic criterion for functions in these classes are given as below(also see [6,19,20,24]).

For and a function is said to be in the class

(i) uniformly starlike functions of order α is denoted by if it satisfies the condition

[picture] | (4)

and

(ii) uniformly convex functions of order α denoted by if it satisfies the condition

[picture] | (5)

Indeed it follows from (4) and (5) that

[picture] | (6)

Remark 1. It is of interest to note that and

Motivated by above definitions we define the following subclasses of due to Murugusundaramoorthy and Magesh [16].

For and we let be the subclass of consisting of functions of the form (1) and satisfying the analytic criterion

[picture] | (7)

and also, let be the subclass of consisting of functions of the form (1) and satisfying the analytic criterion

[picture] | (8)

We further let and

Suitably specializing the parameters we note that

1. [6]

2. [24]

3. [6]

4. [1]

5. [21]

6. [6]

7. [23]

8. [6]

9. [1]

10. [21].

Now we state the following characterization properties for the classes and due to Murugusundaramoorthy and Magesh [16].

Theorem 1. A function of the form (1) is in if

[picture] | (9)

Theorem 2. A function of the form (2) is in if and only if

[picture] | (10)

Theorem 3. A function of the form (1) is in if

[picture] | (11)

Theorem 4. A function of the form (2) is in if and only if

[picture] | (12)

We recall here a generalized Bessel function defined in [2] and given by

[picture] | (13)

which is the particular solution of the second order linear homogeneous differential equation

[picture] | (14)

where which is natural generalization of Bessel’s equation. The differential equation (14) permits the study of Bessel function, modified Bessel function, spherical Bessel function and modified spherical Bessel functions all together. Solutions of (14) are referred to as the generalized Bessel function of order p. The particular solution given by (13) is called the generalized Bessel function of the first kind of order p. Although the series defined above is convergent everywhere, the function is generally not univalent in . It is of interest to note that when we re-obtain the Bessel function of the first kind and for the function becomes the modified Bessel function Now, we consider the function defined by the transformation

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

we can express as

[picture] | (15)

where This function is analytic on and satisfies the second-order linear differential equation

Now, we considered the linear operator

defined by

[picture] | (16)

where . For convenience throughout in the sequel, we use the following notations

and for if let,

[picture] | (17)

The generalized Bessel function is a recent topic of study in Geometric Function Theory (e.g. see the work of [2,3,4,5] and [14]. Motivated by results on connections between various subclasses of analytic univalent functions by using hypergeometric functions (see [8,13,15,22,25]) and by work of Baricz [2,3,4,5], we obtain necessary and sufficient condition for the function to belong to the classes and

2. Main Results and Their Consequences

Lemma 1. [5] If then the function satisfies the recursive relation

for all

Theorem 5. If and then if and only if

[picture] | (18)

Proof. Since

[picture] | (19)

according to Theorem 2, we must show that

[picture] | (20)

Now,

But the last expression is bounded above by if and only if (18) holds.

Remark 2. In particular when and the condition 18 becomes

[picture] | (21)

which is necessary and sufficient condition for to be in where

Theorem 6. If and then is in if and only if

[picture] | (22)

Proof. In view of Theorem 4,we need to show that

Now

By a simplification, we see that the last expression is bounded above by if and only if (22) holds.

3. Inclusion Properties

A function is said to be in the class , , if it satisfies the inequality

The class was introduced earlier by Dixit and Pal [9]. If we put

we obtain the class of functions satisfying the inequality

which was studied by (among others) Padmanabhan [17] and Caplinger and Causey [7]. Making use of the following lemma, we will study the action of the Bessel function on the classes

Lemma 2. [9] If is of form (1), then

[picture] | (23)

The result is sharp.

Theorem 7. Let If and If and if the inequality

[picture] | (24)

is satisfied, then .

Proof. Let f be of the form (1) belong to the class By virtue of Theorem 4, it suffices to show that

Since then by Lemma 2 we have Hence

[picture] | (25)

Further, proceeding as in Theorem 5

But this last expression is bounded above by if and only if (23) holds.

Theorem 8. Let and then

is in if and only if inequality

[picture] | (26)

Proof. Since

then by Theorem 4 we need only to show that

Now

Further, proceeding as in Theorem 5

which is bounded above by if and only if (26) holds.

Concluding Remarks

If we put c=-1 and b=1 in above theorems we obtain analogous results of (21). Further by taking β=0 and specializing the parameter λ we can state various interesting results (as proved in above theorems) for the various subclasses listed in the introduction.

Acknowledgement

The authors thank the referees for their valuable suggestions to improve the paper in present form.

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