A Generalized Subclass of p-VALENT Analytic Functions

10

V

https://doi.org/10.22214/ijraset.2022.43497

May 2022

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue V May 2022- Available at www.ijraset.com

A Generalized Subclass of p-VALENT Analytic Functions Jitendra Awasthi Department of Mathematics, S.J.N.P.G. College, Lucknow-226001 Abstract: In this paper, a class  p , ( ,  ,  , k ) of functions F analytic in the unit disk U={z:|z|<1} of the form 

F ( z)  z p   a p t z p t , (p, n  N, a p t  0 for t 

n) is considered. Coefficient inequality, distortion theorem, extreme

t n

points, starlikeness and convexity for this class are obtained.

Keywords: Analytic functions, p-valent functions, extreme points, radii of starlikeness and convexity.

2010 AMS Subject classification: Primary 30C45.

Let  p , n

I. INTRODUCTION denotes the class of p-valent analytic functions in the unit disk U={z:|z|<1} which are of the form 

(1.1)

F ( z)  z p   a p t z p t , p,n  N, a p t  0 . t n

n

A function P ( z )  p  p n z  ....., ( n  1) analytic in U is said to be in  p ,n ( ) if (1.2)

P ( z )  p < ( p   ) , 0   < p, z  U .

Alternatively, in terms of subordination, it is said that P(z) is in  p ,n ( ) if (1.3)

P( z)  p  ( p   ) z

where ‘’ stands for subordination which is defined by saying that F is subordinate to G written as F  G if F(z) = G(ɸ(z)), zU for some analytic functions ɸ (z) such that ɸ (0) = 0 and | ɸ (z)| < 1 for zU. Now, a class criterion Mp,n(,,,k) whose members P(z) = p + pnzn … (n1), analytic in U satisfy the condition (1.4)

P( z) 

k  ( k  2  )  z 1  ( 2   1)  z

with 1  2 2, 2< k  p, 0 < 1 and 0   <

p (p  k)  2 

is considered.

Equivalently (1.4) can be written as: (1.5)

P( z)  k  (2  1) P( z)  (k  2 )

Note that Mp,n(,1,1/2,p) Mp,n() and Mp,n(,,,p) Mp,n(,,).

On putting P(z) =

F ' ( z) z p 1

in (1.5) it follows that

©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |

4751

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue V May 2022- Available at www.ijraset.com

F ' ( z) k z p 1 , F ' ( z) (2  1) p1  (k  2 ) z

(1.6)

F(z)  p , n .

The class of such functions F(z) satisfying (1.6) is denoted by  p , ( ,  ,  , k ) and

 p,n(, , , p)   p,n(, , ).

Clearly, these classes are more general than the classes studied by Gupta and Jain [2,3], Aouf [1], Juneja–Mogra [4], Thirupathi Reddy [6] and Kulkarni, Aouf, Joshi [5] etc.

In this paper, coefficient inequalities, distortion theorem, closure theorem, radii of starlikeness and convexity for the class  p,n(, , , k) are obtained.

II. COEFFICIENT INEQUALITIES: Theorem 2.1: A function F(z)Tp,n is in the class  p,n(, , , k) if and only if 

 ( p  t )a

(2.1)

p t



tn

{2 ( p   )  ( p  k )(1   )} .

1  ( 2  1)

The result is sharp, the extremal function being

F (z)  z p 

(2.2)

Proof: Let F(z) 

p,n(,

{2  ( p   )  ( p  k )(1   )} p  t z ( p  t ){1  ( 2   1)  }

for t  n.

, , k), then

F ' ( z) k z p1 F ' ( z) (2  1) p 1  (k  2 ) z 

( p  k )   ( p  t )a pt z t tn



.



{(2  1)p  ( k  2)}  ( 2  1) ( p  t )z

t

tn –

On letting z  1 through real values, it gives 

( p  k )   ( p  t )a p  t t n 

 {( 2  1) p  ( k  2 )}   ( 2  1)  ( p  t )a p  t tn

or, 

 ( p  t )a tn

p t



2 ( p   )  ( p  k )(1   ) .

1  ( 2  1)

Conversely, let (2.1) holds, then for |z| = 1

| F ' ( z ) z 1 p  k |   | ( 2   1) F ' ( z ) z 1 p  ( k  2  ) |

©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |

4752

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue V May 2022- Available at www.ijraset.com 

 ( p  k )   ( p  t )a p t tn 

 {2  1) p  ( k  2 )}   ( 2  1)  ( p  t )a p  t t n 

 {1  {( 2  1)} ( p  t )a p  t  2 ( p   )  ( p  k )(1   ) tn

 0.

The result is sharp for function given by (2.2).

Corollary 2.2: For k = p, F(z)  p,n(, , , p) if and only if 

 ( p  t )a

p t



tn

2 ( p   ) p , 0 , 1   ( 2  1) 2 0 < 1, 1  2 2.

For n = 1, the result of Kulkarni et.al. [5] follows.

Corollary 2.3: For k = p and  = 1, F(z)  p,n(, , 1, p) if and only if 

 ( p  t )a

p t



tn

2 ( p   ) , 0  < p, 0 <  1.

1 

For n = 1,  = 1, the result of Juneja and Mogra [4] follows.

For n = 1, p = 1, the result of Gupta and Jain [3] follows.

Corollary 2.4: For k = p,



1 1   , F ( z )   p ,n   ,  , , p  , if and only if 2 2  



 ( p  t )a tn

pt



(1  ) ( p   ) p , 0 1   2

.

For n = 1, the result of Thirupathi Reddy [6] follows.

III. DISTORTION THEOREM Theorem 3.1: If F(z)  p,n(,,,k), then for |z| = r < 1 (3.1)

 2 ( p   )  ( p  k )(1   )  pn rp  r | F ( z) |  ( p  n){1   (2  1)}   2(p  )  (p  k )(1  )  pn  rp   r .  (p  n){1  (2  1)} 

and (3.2)

 2 ( p   )  ( p  k )(1   )  pn1 pr p1   | F ' ( z) | r { 1   ( 2   1 )}    2(p  )  (p  k)(1  )  pn1  pr p1   . r { 1   ( 2   1 )}  

©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |

4753

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue V May 2022- Available at www.ijraset.com Proof: From Theorem 2.1, it follows that 

(3.3)

a t n

p t

 2(p  )  (p  k)(1  )   .  (p  n){1  (2  1)} 

Hence 

| F ( z ) | r   a p  t r p  t p

t n

 2(p  )  (p  k )(1  )  pn  rp   r  (p  n){1  (2  1)}  and 

| F ( z ) | r p   a p  t r p  t t n

 2(p  )  (p  k )(1  )  pn  rp   r .  (p  n){1  (2  1)}  Hence (3.1) follows.

In the same way, it follows that 

| F ' ( z ) | pr p 1   ( p  t ) a p  t r p  t 1 t n

 2(p  )  (p  k )(1  )  pn 1  pr p1   r {1  (2  1)}   and 

| F ' ( z ) | pr p 1   ( p  t ) a p  t r p  t 1 tn

 2(p  )  (p  k )(1  )  pn 1  pr p1   . r { 1   ( 2   1 )}   This completes the proof of the theorem.

The above bounds are sharp. Equalities can be attained for the function (3.4)

 2 ( p   )  ( p  k )(1   )  pn F ( z)  z p    z , z  r .  ( p  n){1   (2  1)}  IV.

CLOSURE THEOREMS 

Theorem 4.1: If F(z) 

p,n(,

, , k) and

G ( z )  z p   b p t z pt

are also in

tn

p  p,n(,,,k), then H ( z )  z 

1  ( a p t  b p t ) z p t  2 t n

Proof: Since F(z) and G(z) both belong to 

p,n(,

is also in 

p,n(,,,k).

, , k), then from Theorem 2.1

©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |

4754

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue V May 2022- Available at www.ijraset.com 

(4.1)

{1  ( 2  1)} ( p  t )a p  t  2( p   )  ( p  k )(1   ) . t n

and 

(4.2)

{1  ( 2  1)} ( p  t ) b p  t  2 ( p   )  ( p  k )(1   ) . tn

So for H(z), it follows that  1 {1  ( 2  1)} ( p  t )( a p  t  b p  t ) 2 tn

 2(p  )  (p  k )(1  ) .

Using (4.1) and (4.2). Therefore, H(z) 

p,n(,

, , k).

V.

EXTREME POINTS

p

Theorem 5.1: Let F n 1 ( z )  z and

 2 ( p   )  ( p  k )(1   )  p t Ft ( z )  z p    z , t n then F(z)  p,n(,,,k) if and only if it can be expressed in  ( p  t ){1   (2  1)}  the form 

F ( z) 



 t Ft ( z ) , where t  0 and

t  n 1



t

 1.

t  n 1

Proof: Suppose 

F ( z) 



 2 ( p   )  ( p  k )(1   )  pt t z , ( p  t ){1   (2  1)}  t n 

p

  F (z) z    t

t

t n1

then 

 (p  t) tn

{2( p   )  ( p  k )(1   )} t ( p  t ){1  ( 2  1)}



{2( p   )  ( p  k )(1  )} .

1  ( 2  1)

Thus, by Theorem 2.1, F(z)  p,n(, , , k).

Conversely, suppose F(z)  p,n(, , , k). Hence, by Theorem 2.1, it follows that

a p t 

2( p  )  ( p  k )(1  ) , tn ( p  t ){1  ( 2  1)}

Setting

n 

( p  n ){1  ( 2  1)} a p n , n  1,2,...

2( p   )  ( p  k )(1  )

and 

 n 1  1    n . tn

It follows that

©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |

4755

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue V May 2022- Available at www.ijraset.com 

F ( z) 

  F ( z ). t

t

t  n 1

This completes the proof.

The extreme points for the class 

F n 1 ( z )  z

p,n(,

, , k) are given by

p

and

 2  ( p   )  ( p  k )(1   )  p  t Ft ( z )  z p    z , t n . ( p  t ){ 1   ( 2   1 )}   VI. RADIUS OF STARLIKENESS: Theorem 6.1: If F(z)  p,n(, , , k), then the function F(z) is startlike in the disk 0 < |z| < r = r(, , , k, n) where 1/ n

  p{1  ( 2  1)} r ( , , , k, n )  inf   nN 2 ( p   )  ( p  k )(1   )  

,nN

Proof: It is enough to show that

zF ' ( z )  p  p for F ( z)

|z| < 1.

or, 

zF ' ( z ) p  F (z)

  t a p t z t t n 

1   a p t z

p t

t n

or, 

  t t t a | z |  p 1  a z   p t p  t   tn  tn 

or, 

pt  a p t | z |t  1 . tn  p 

  But, Theorem 2.1 gives 

 (p  t) a tn

pt



2 ( p   )  ( p  k )(1   ) 1   ( 2  1)

.

Thus, F(z) is starlike if

  p[1  ( 2  1)] | z |    2( p   )  ( p  k )(1  ) 

1/ n

, n  1,2,...

.

©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |

4756

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue V May 2022- Available at www.ijraset.com VII. RADIUS OF CONVEXITY Theorem 7.1:If F(z)  p,n(, , , k), then F(z) is convex in the disk 0 < |z| < r = r(, , , k, n), where

  p 2 {1  ( 2  1)} r ( , , , k , n )  inf   nN ( p  n )[ 2 ( p   )  ( p  k )(1   )]  

1/ n

n  1,2.. .

Proof: Putting zF '(z) in place of F(z) in Theorem 6.1, the result follows.

REFERENCES [1] [2] [3] [4] [5] [6]

Aouf, M.K., Certain classes of p-valent functions with negative coefficients II, Indian J. Pure Appl. 19(8) (1988), 761-767.

Gupta, V.P. and Jain, P.K., Certain classes of univalent functions with negative coefficients, Bull. Austr. Math. Soc. Vol. 14(1976), 409-416.

Gupta, V.P. and Jain, P.K., Certain classes of univalent functions with negative coefficients, Bull. Austr. Math. Soc. Vol. 15(1976), 467-473.

Juneja, O.P. and Mogra, M.L., Radii of convexity for certain classes of univalent analytic functions, Pacific Jour. Math. 78, (1978), 359–368.

Kulkarni, S.R., Aouf, M.K. and Joshi, S.B., On a subfamily of p–valent functions with negative coefficients, MATEMATИЧҠИ BECHИҠ 46(1994), 71–75.

Thirupathi Reddy, P., Certain classes of p–valent functions with negative coffocients, Indian J. Pure Appl. Math. 430 (2001), 203—206.

©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |

4757