10
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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)), zU for some analytic functions ɸ (z) such that ɸ (0) = 0 and | ɸ (z)| < 1 for zU. Now, a class criterion Mp,n(,,,k) whose members P(z) = p + pnzn … (n1), 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
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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) p1 (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
tn
{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 p1 F ' ( z) (2 1) p 1 (k 2 ) z
( p k ) ( p t )a pt z t tn
.
{(2 1)p ( k 2)} ( 2 1) ( p t )z
t
tn –
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 tn
or,
( p t )a tn
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 ) |
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( p k ) ( p t )a p t tn
{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 ) tn
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
tn
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
tn
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 tn
pt
(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 ) pn rp r | F ( z) | ( p n){1 (2 1)} 2(p ) (p k )(1 ) pn rp r . (p n){1 (2 1)}
and (3.2)
2 ( p ) ( p k )(1 ) pn1 pr p1 | F ' ( z) | r { 1 ( 2 1 )} 2(p ) (p k)(1 ) pn1 pr p1 . r { 1 ( 2 1 )}
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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 ) pn 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 ) pn 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 ) pn 1 pr p1 r {1 (2 1)} and
| F ' ( z ) | pr p 1 ( p t ) a p t r p t 1 tn
2(p ) (p k )(1 ) pn 1 pr p1 . 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 ) pn 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 pt
are also in
tn
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
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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 ) . tn
So for H(z), it follows that 1 {1 ( 2 1)} ( p t )( a p t b p t ) 2 tn
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 ) pt t z , ( p t ){1 (2 1)} t n
p
F (z) z t
t
t n1
then
(p t) tn
{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 ) , tn ( 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 . tn
It follows that
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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 nN 2 ( p ) ( p k )(1 )
,nN
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 tn tn
or,
pt a p t | z |t 1 . tn p
But, Theorem 2.1 gives
(p t) a tn
pt
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,...
.
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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 nN ( 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.
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