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A Subclass of 1Valent and Analytic Functions Associated With Dzoik Srivastava Linear Operator

 

Chena Ram*, Saroj Solanki#

Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur, Rajasthan, INDIA.

Corresponding Addresses:

*[email protected], #[email protected]

Research Article

 

Abstract:In this paper, a new subclasses 1 of analytic and 1valent functions has been introduced. Necessary and sufficient conditions for these classes are discussed. Distortion properties and the result of modified Hadamard product are obtained for the subclasses1 Some other results for the same subclasses of functions are also obtained.

Key words: Hadamard product, Dzoik -Srivastava operator, generalized fractional operator, 1valent functions, analytic functions, linear operator, distortion theorem, Riemann-Lioueville operator.


 

1. Introduction

Let 1 denote the class of functions of the form

1

which are analytic and 1valent  in the unit disk1

Function 1 is given by

1

then the Hadamard product (or Convolution) of  1 and  1 is defined by

1

Let 1 denote the subclass of  1 consisting of functions of the form

1

The Generalized hypergeometric function q1s1 for positive real values of 1 and 1 (1) is defined by

q1s1 q1s1=  11        (1.5)

1

where  1 is the pochhammer symbol defined by

1

Corresponding to a function 1 is defined by 1q1s1         (1.7)

The Dziok and Srivastava operator [6]  1 is defined by

1

1          (1.8)

1

If we set  1, then the linear operator 1 reduces to linear operator 1 as

 

                1    (1.10)    

If we put  1 in above equation (1.10), then it reduces to Carlson-Shaffer operator [2] as

1

In particular, if we put 1, then it reduces to Ruscheweyh operator [10] given by
1

If we set 1, then (1.10) reduces to Bernadi-Libera-Livingston integral operator given earlier by (see [11, 1, 8]) as

1

Now, we recall the following definition of fractional derivative operator due to Owa [9].

Definition1. The fractional integral of order 1 for function 1 is defined by

1

where the analytic function 1 is defined in a simply-connected region of the 1 -plane containing the origin,  and multiplicity of 1 is removed by requiring  1to be real when 10.

The fractional derivative operator of order1for an analytic function 1 is defined by

1

where the conditions, under which (1.15) is valid, are similar to those stated with (1.14).

Definition 2. Under the hypotheses of (1.15), the fractional derivative of function 1order1 is defined by

1

In particular case, if we let  1 then (1.10) reduces to linear operator 1due to Srivastava and Owa [3] is defined by

1

where 1.

For details, one can see [4, 5].

Definition 3. A function 1defined by (1.4) is said to be in the class1if it satisfied the following relations (cf. [7])
           

 
1

1

A function 1defined by (1.4) is said to be in the class1if and only if

1

 

2. Coefficient Estimates

Theorem 2.1. Let the function 1 be defined by (1.4). Then 1 if and

only if

1

The result is sharp for the function

1

where 1is defined by (1.9).

Proof. Let 1Then in view of (1.18), we have

1

1

by using 1in (2.3), we get

1

taking values of1on the real axis and let 1through real values then

1

1

Conversely, let inequality (2.3) hold true, then

1

1

1

1

By maximum modulus principle, this implies that 1

1

The result is sharp for the functions

1

where 1is defined by (1.9).

Theorem 2.2. Let the function 1 be defined by (1.4). Then 1 if and

only if

1

The result is sharp for the function

1

where 1is defined by (1.9).

Proof. On using (1.18) and (1.19), we easily arrive at the desired result (2.4) and (2.5).

 

3. Closure theorem

Let the function 1be defined for 1 by

1

Theorem 3.1. Let the function 1 defined by (3.1) be in the class1 for each

1. Then the function1 defined by

1

1

Proof.1 then by using (2.1)

1

where 1is given by (1.9). Therefore

1

1

which shows that 1

The theorem is completely proved.

Theorem 3.2. Let the function 1 defined by (3.1) be in the class1 for each

1. Then the function1 defined by

1

1

Proof. The proof follows exactly on the same lines as that of Theorem 3.1.

4. Distortion theorem for the classes 1 and 1 

Theorem 4.1. Let the function 1 defined by (1.4) be in the class1. Then

1

and

1

for  1provided that 1 where 1is defined by

(1.9).

Proof. by using(1.4), then we have

1

1

by using (2.1), we get

1

here 1is defined by (1.9).

We know that 1is non decreasing for1, then we have

1     (4.3)

Then,

1

and

1

 

The theorem is completely proved.

 

Corollary 4.2. Under the hypothesis of theorem (4.1), 1is included in a disc with it’s

centre at the origin and radius 1 given by

1

Theorem 4.3. Let the function 1 defined by (1.4) be in the class1. Then

1

and

1

for  1provided that 1

Where 1 is defined by (1.9).

Proof. On using (1.4) and (2.4), we easily arrive at the desired result (4.7) and (4.8).

Corollary 4.4. Under the hypothesis of theorem (4.2), 1is included in a disc with it’s

centre at the origin and radius 1 given by

1

Theorem 4.5. Let the function 1 defined by (1.4) be in the class1. Then

1

and

1

Proof. By using (1.8), we get

1 1,     (4.12)

1

           
by using (2.1), (4.12) reduces to

1

            and

1

This completes the proof of theorem 4.5.

Theorem 4.6. Let the function 1 defined by (1.4) be in the class1. Then

1

and

1

Proof. The proof follows exactly on the same lines as that of Theorem 5.1.

Corollary 4.7.  Let the function 1 defined by (1.4) be in the class1 and let

1 in (4.10) and (4.11) then it reduces to (4.15) and (4.16)

respectively given by

1

and
1

Corollary 4.8.  Let the function 1 defined by (1.4) be in the class1 and let

1 in (4.10) and (4.11) then it reduces to (4.17) and (4.18)

respectively given by

1

and
1

Corollary 4.9.  Let the function 1 defined by (1.4) be in the class1 and let

1 in (4.13) and (4.14) then it reduces to (4.19) and (4.20)

respectively given by

1

and

1

Corollary 4.10.  Let the function 1 defined by (1.4) be in the class1 and let

1 in (4.13) and (4.14) then it reduces to (4.21) and (4.22)

respectively given by

1

and  
            
1

 

 

References

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