* IJSRR, 8(2) April. – June., 2019 * Page 1

** Research article Available online www.ijsrr.org **

** ISSN: 2279–0543 **

**International Journal of Scientific Research and Reviews**

**International Journal of Scientific Research and Reviews**

**Certain Subclass of Analytic Univalent Functions Using q - Differential **

**Operator **

**G. Thirupathi **

Department of Mathematics, Ayya Nadar Janaki Ammal College, Sivakasi - 626 124, Tamilnadu, India. Email: gtvenkat79@gmail.com.

**ABSTRACT **

In this paper, we define a new subclass of analytic univalent function using q - differential

operator, which generalizes Ruschewayh differential operators. Coefficient inequalities,

Subordination, extreme points and integral means inequalities results are obtained.

**KEYWORDS:**

Univalent, subordinating factor sequence, integral means, Hadamard product, q-
derivative operator, subordination.

***Corresponding author: **

**G. Thirupathi **

Department of Mathematics,

Ayya Nadar Janaki Ammal College,

Sivakasi - 626 124, Tamilnadu, India.

* IJSRR, 8(2) April. – June., 2019 * Page 2

**INTRODUCTION, DEFINITIONS AND PRELIMINARIES **

Let us denote U{*z* \∣ ∣*z* 1} and let A denote the class of functions of the form

2 3

2 3

( ) , * _{n}* 0,

*f z* *z* *a z* *a z* *a* (1.1)
which are analytic in the open disc U.

Mohammed and Darus1 studied approximation and geometric properties of the *q*operators
in some subclasses of analytic functions in compact disk. Recently, Purohit and Raina2,3 have used
the fractional *q*calculus operators in investigating certain classes of functions which are analytic in
the open disk. Also Purohit4 studied these *q* operators, defined by using the convolution of
normalized analytic functions and *q*hypergeometric functions.

The *q*derivative operator of a function *f* is defined by
( ) ( ) ( ) ( 0)

(1 )
*q*

*f z* *f qz*

*D f z* *z*

*q z*

(1.2)
and (*D f _{q}* )(0)

*f*(0),provided that the function

*f*is differentiable at 0.We note that

( ) ( )
*q*

*D f z* *f z* as *q*_{}1

.

Also, from [1.2], we have *D*_

_{ }

*q f z*( )=1+

2

[ ] _{n}*n*
*n*

*n a z*

_{}

, where
[ ] 1 . 1

*n*

*q*
*n*

*q*

(1.3)

The Hadamard product of two functions

2

( ) _{n}*n*

*n*

*f z* *z* *a z*

_{}

and
2

( ) _{n}*n*

*n*

*g z* *z* *b z*

_{}

is given by

2

( * )( ) _{n}_{n}*n*.
*n*

*f* *g z* *z* *a b z*

_{}

(1.4)
Recently, Kanas and Raducanu 5, defined and investigated Ruschewayh *q*differential operator as follows:

For *f*A , generalized Ruschewayh *q*differential operator is defined by

###

###

2

( )

( ) , .

( 1)! (1 )

*q* *n*

*q* *n*

*n* *q*

*n*

*R f z* *z* *a z* *z*

*n*

** **

**

###

U (1.5)Here *R f z _{q}*0 ( )

*f z R f z*( ),

*1 ( )*

_{q}*zD f z*( ) and

_{q}1

1 ( ( ))

( )

[ ]!
*m* *m*
*q*
*q*

*zD* *z* *f z*
*R f z*

*m*

.

It can be seen that if we let*q*1, then *R f z _{q}*

###

reduces to the well-known Ruschewayh differential* IJSRR, 8(2) April. – June., 2019 * Page 3

*D _{q}*

###

*R f z*( )

_{q}###

= 1+2

*n*

###

[n]**( , )

_{q}*n*

*n*1

*n*

*a z* (1.6)

where

( , ) ( ) . ( 1)! (1 )

*q*
*q*

*q*

*n*
*n*

*n*

**

** **

**

(1.7)

Using the generalized Ruschewayh *q*differential operator, we define the following class
( , , ).

*q* *n *
S

**Definition 1.1: **Let S* _{q}*( , , )

*n *be the class of functions

*f*A satisfying

###

###

###

###

###

###

*q* *q*

*q* *q*

*D* *R f z*
*Re*

*D* *R f z*

**

** **

(1.8)

for some 0

_{ }

1 1 1

0**1,** _{} _{} 0 ,** _{} .

**Definition 1.2: **(Subordination) Given two functions *f z*

_{ }

and*g z*

_{ }

, which are analytic in U.Then
we say that the function ( )*f z* is subordinate to ( )*g z* in U, if there exists an analytic function ( )*w z* in
Usuch that (0)*w* 0, | ( ) | 1(*w z* *z*U such that ( )) *f z* *g w z*( ( )), denoted by ( )*f z* *g z*( ).

**Definition 1.3: **(Subordinating Factor Sequence) A sequence

###

1

*n* _{n}

*b*

of complex numbers is said to

be a subordinating sequence if, whenever

1

( ) *n*

*n*
*n*

*f z* *a z*

_{}

, *a*

_{1}1 is regular, univalent and convex in U, we have

1

( ), ( ).

*n*
*n* *n*
*n*

*b a z* *f z* *z*

###

U (1.9)Motivated by the concept introduced by Serap Bulut7, Selvaraj8, in this paper, we obtain coefficient bounds, extreme points and integral means inequalities for the above said function class.

Let T denote the subclass of *f* A consisting of functions of the form

2

( ) *n*

*n*
*n*

*f z* *z* *a z*

_{}

(1.10)
**COEFFICIENT INEQUALITIES **

**Theorem 2.1: **Let *f z*( )A of the form [1.1]. If the inequality

###

###

2

( , , ) 2(1 ), ,

*n* *n*

*n*

*a* *z*

* * **

###

B U (2.1)* IJSRR, 8(2) April. – June., 2019 * Page 4

###

( , , ) ( , ) (1 ) ( , ) ( , ) (1 ) ( , )
*n* * * *n* _{}*q* *n* * q* *n* *q* *n* * q* *n* ** _{}

B (2.2)

and * _{q}*( , )

*n*is given by (1.7) then

*f*S

*( , , )*

_{q}*n *.

*Proof: *Suppose that the inequality (2.1) holds. Then for*z*U, we define the function *F* by

###

###

###

###

###

###

( ) *q* *q* .

*q* *q*

*D* *R f z*
*F z*

*D* *R f z*

**

** **

(2.3)

It is sufficient to show that

( ) 1 1,

_{}

_{}

.
( ) 1
*F z*

*z*
*F z*

U (2.4)

Now,

###

###

###

_{}

###

_{}

###

###

###

_{}

###

_{}

(1 ) ( ) 1

( ) 1 (1 )

*q* *q* *q* *q*

*q* *q* *q* *q*

*D* *R f z* *D* *R f z*
*F z*

*F z* *D* *R f z* *D* *R f z*

** **

** **

**
**

###

###

###

###

###

1 2

1 2

1 2

1 2

2

( , ) (1 ) ( , )

(2 ) ( , ) (1 ) ( , )

( , ) (1 ) ( , )

(2 ) ( , ) (1 ) ( , )

( , ) (1 ) ( , )

(2

*n*

*q* *q* *n*

*n*

*n*

*q* *q* *n*

*n*

*n*

*q* *q* *n*

*n*

*n*

*q* *q* *n*

*n*

*q* *q* *n*

*n*

*n* *n* *n* *a z*

*n* *n* *n* *a z*

*n* *n* *n* *a* *z*

*n* *n* *n* *a* *z*

*n* *n* *n* *a*

** ** ** * * **

** ** ** * * **

** ** ** * * **

** ** ** * * **

** ** ** * * **

_{} _{}

_{} _{}

_{} _{}

_{} _{}

###

###

###

###

###

###

2

1
) * _{q}*( , ) (1 )

*( , )*

_{q}

_{n}*n*

*n* *n* *n* *a*

** ** ** * * **

_{}

_{}

_{}

Therefore, *f*S* _{q}*( , , )

*n *.

**INTEGRAL MEANS INEQUALITIES **

* Lemma 1: [Selvaraj et al.*9] If the functions

*f*and

*g*are analytic in U with ( )

*f z*

*g z*( ), then for

**0

*and*

*z*

*rei*(0

*r*1),

2 2

0 *f z*( ) *d* 0 *g z*( ) *d* .

** _{}** _{}

** **

* IJSRR, 8(2) April. – June., 2019 * Page 5
Silverman10 found that the function

2 2( )

2

*z*

*f z* *z* is often extremal over the family T and
applied this function to resolve his integral means inequality, conjectured and settled in11 , that
2 2 _{2}

0 *f z*( ) *d* 0 *f z*( ) *d* .

** ** ** **

** **

###

###

(3.2)for all *f*T . In12, Silverman also proved his conjecture for the subclasses T*( )** and K( )** of T .

**Theorem 3.1: **Suppose *f*S* _{q}*( , , ),

*n *

**0, 0

**1 and

*f z*

_{2}( )is defined by

_{2}2

2

2(1 ) ( )

( , , )

*f z* *z* ** *z*

* *

B , where

2( , , )* * _{}*q*(2, ) (1** * *) *q*(2, )** *q*(2, )** (1* *) *q*(2, )** _{}

B (3.3)

and (2, ) [2] (2 ) . (2 1)! (1 )

*q*
*q*

*q*

**

** **

**

(3.4)

Then for *z**rei*(0 *r* 1),we have

2 2 _{2}

0 *f z*( ) *d* 0 *f z*( ) *d* .

** _{}** _{}

** **

###

###

(3.5)*Proof: *Using (1.10) and (3.5), it is enough to prove that

2 1 2

0 0

2 2

2(1 )

1 1 .

( , , )
*n*

*n*
*n*

*a z* *d* *z d*

**
**

** ** _{}

** **

* *

_{}

###

###

_{B}(3.6)

By Lemma 1, it suffices to show that

1

2 2

2(1 )

1 1 .

( , , )
*n*

*n*
*n*

*a z* ** *z*

* *

_{}

B

Setting 1

2 2

2(1 )

1 1 ( ),

( , , )
*n*

*n*
*n*

*a z* ** *w z*

* *

_{}

B

and using (2.1), we obtain *w z*( ) is analytic inU, (0)*w* 0, and
2 1

2 2

( , , ) ( , , )

( ) ,

2(1 ) 2(1 )

*n* *n*

*n* *n*

*n* *n*

*w z* * * *a z* *z* * * *a* *z*

** **

###

B###

Bwhere B* _{n}*( , , )

* *is given by (2.2). This completes the proof of the theorem.

**SUBORDINATION RESULTS **

* IJSRR, 8(2) April. – June., 2019 * Page 6

**Lemma 2: The sequence**_{ }

1

*n* _{n}

*b*

is a subordinating factor sequence if and only if

1

1 2 _{n}*n* 0, ( ).

*n*

*Re* *b z* *z*

###

U (4.1)

**Theorem 4.1: **Let *f*S* _{q}*( , , )

*n *and ( )

*g z*be any function in the usual class of convex functions K, then

###

###

2 2 ( , , ) ( * )( ) ( ) ( )2 2(1 ) ( , , ) *f* *g z* *g z* *z*

* *

** * *

U

B

B (4.2)

where 0**1with * _{q}*( , )

*n*given by (3.4) and

###

###

###

2###

2

2(1 ) ( , , )

( ) ( ).

( , , )

*Re f z* ** * * *z*

* *

B U

B (4.3)

*Proof: *Let *f*S* _{q}*( , , )

*n *and suppose that

2

( ) _{n}*n* .

*n*

*g z* *z* *b z*

_{}

K
Then, for *f*A given by (1.1), we have

###

###

###

###

2 2 2 2 2 ( , , ) ( , , ) ( * )( ) .2 2(1 ) ( , , ) 2 2(1 ) ( , , )

*n*
*n n*
*n*

*f* *g z* *z* *a b z*

* * * *
** * * ** * *
_{} _{}
_{}

###

_{}B B

B B (4.4)

Thus, by Definition 3, the subordination result holds true if

###

###

2

2 _{1}

( , , ) 2 2(1 ) ( , , )

*n*
* *
** * *
_{} _{}
B
B

is a subordinating factor sequence, with*a*_{1}1. In view of Lemma 2, this is equivalent to the
following inequality

###

###

2 1 2 ( , , )1 0, ( ).

2(1 ) ( , , )
*n*
*n*
*n*

*Re* * * *a z* *z*

** * *
_{} _{}

###

U BB (4.5)

Now, for *z* *r* 1, we have

###

###

###

###

###

###

2 1 2 2 1 2 2 2 ( , , ) 12(1 ) ( , , )

( , , ) ( , , )

1

2(1 ) ( , , ) 2(1 ) ( , , )

*n*
*n*
*n*
*n*
*n*
*n*

*Re* *a z*

* IJSRR, 8(2) April. – June., 2019 * Page 7

###

###

###

###

###

###

###

###

###

###

2 1 2

2 2

2

2 2

( , , ) ( , , )

1

2(1 ) ( , , ) 2(1 ) ( , , )

( , , ) 2(1 )

1

2(1 ) ( , , ) 2(1 ) ( , , )

1 0, 1

*n*
*n*
*n*

*a r*
*r*

*r* *r*

*r* *z* *r*

* *
* *

** * * ** * *

* * **

** * * ** * *

###

BB

B B

B

B B

where we have also made use of the assertion (2.1) of Theorem 2.1. This evidently proves the

inequality (4.5) and hence also the subordination result (4.4) asserted by Theorem 4.1. The

inequality (4.3) asserted by Theorem 4.1 would follow from (4.2) upon setting

1

( ) .

1

*j*

*j*

*z*

*g z* *z*

*z*

###

KFinally, we consider the function ( )*q z* is given by

2 2

2(1 ) ( )

( , , )

*q z* *z* ** *z*

* *

B (4.7)

and * _{q}*(2, )

**is given by (3.4). Clearly

*q*S

*( , , )*

_{q}*n *. For this function (4.2) becomes

###

###

2 2

( , , )

( ) .

2 2(1 ) ( , , ) 1

*z*
*q z*

*z*

* *

** * *

B B

Moreover, it can easily be verified for the function ( )*q z* given by (4.7) that

###

###

2 2

( , , ) 1

( ) ( ),

2 2(1 ) ( , , ) 2

*min Re* * * *q z* *z*

** * *

_{} _{}

U B

B

which evidently completes the proof of Theorem 4.1.

**EXTREME POINTS **

**Theorem 5.1: **Let *f z*_{1}( )*z* and

( ) 2(1 ) ( 2,3, ) ( , , )

*k*
*k*

*k*

*f z* *z* ** *z* *k*

* *

B (5.1)

where B* _{k}*( , , )

* *given by (2.2). Then

*f*S

*( , , )*

_{q}*n *if and only if it can be expressed in the form

1

( ) _{k}* _{k}*( ),

*k*

*f z* ** *f z*

_{}

(5.2)
where * _{k}* 0 and

1

1
*k*
*k*

**

###

.* IJSRR, 8(2) April. – June., 2019 * Page 8

1 1

2

1 2

1 2 2

( ) ( ) ( ),

2(1 ) ( , , )

2(1 ) 2(1 )

.

( , , ) ( , , )

*k* *k*
*k*

*k*
*k*

*k* *k*

*k* *k*

*k* *k* *k*

*k* *k* *k* *k* *k*

*f z* *f z* *f z*

*z* *z* *z*

*z* *z* *z* *z*

** **

**

** **

* *

** **

** ** **

* * * *

_{} _{}

_{} _{} _{} _{} _{} _{}

###

###

###

###

###

B

B B

(5.3)

Thus

_{1}

2 2

2(1 )

( , , ) 2(1 ) 2(1 )(1 ) 2(1 ). ( , , )

*k* *k* *k*

*k* *k* *k*

**

** * * ** ** ** ** **

* *

###

B###

B (5.4)

Therefore, we have *f*S* _{q}*( , , )

*n *.

Conversely, suppose that *f* S* _{q}*( , , )

*n *. Since 2(1 ) , ( 2,3, ), ( , , )

*k*
*k*

*a* ** *k*

* *

B

we can set ( , , ) , ( 2,3, ) 2(1 )

*k*

*k* *ak* *k*

* *
**

**

B

and _{1}

2

1 * _{k}*.

*k*

** **

_{}

Now

2 1 2

1 2

1 1

2 1

2(1 ) ( )

( , , )

2(1 ) ( , , )

( ) ( ) ( ).

*k* *k*

*k* *k* *k*

*k* *k* *k* *k*

*k*
*k*

*k* *k*

*k* *k* *k* *k*

*k* *k*

*f z* *z* *a z* *z* *z*

*z* *z* *z*

*f z* *f z* *f z*

**

** **

* *
**

** **

* *

** ** **

_{} _{} _{} _{}

_{} _{}

###

###

###

###

###

###

B

B

This completes the proof of the Theorem 5.1.

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