 Open Access
 Total Downloads : 518
 Authors : T.N. Shanmugam , J.Lourthu Mary
 Paper ID : IJERTV1IS10039
 Volume & Issue : Volume 01, Issue 10 (December 2012)
 Published (First Online): 28122012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Properties Of Universally Prestarlike Functions
Properties Of Universally Prestarlike Functions

Universally prestarlike functions of order 1 in the slit domain
= C \ [1, ) have been recently introduced by S. Ruscheweyh.This notion generalizes the corresponding one for functions in the unit disk (and other
circular domains in C). In this paper, we obtain properties of universally prestarlike functions of order .
Mathematics Subject Classification: 26A33 (main), 33C44
Key Words and Phrases: Prestarlike functions, Universally Prestarlike functions, Properties,etc.,
Let H() denote the set of all analytic functions defined in a domain . For domain containing the origin H0() stands for the set of all function f H() with f (0) = 1. We also use the notation
H1() = {zf : f H0()} . In the special case when is the open unit disk
= {z C : z < 1} , we use the abbreviation H, H0 and H1 respectively for H(), H0() and H1(). A function f H1 is called starlike of order with (0 < 1) satisfying the inequality
( 1
zf ,(z) > (z ) (1.1)
f (z)
and the set of all such functions is denoted by S. The convolution or
Hadamard Product of two functions f (z) = anzn and g(z) = bnzn
n=0
n=0
n=0
n=0
is defined as
n
(f g)(z) = anbnz .
n=0
A function f H1 is called prestarlike of order if
z
(1 z)22 f (z) S (1.2)
The set of all such functions is denoted by R. The notion of prestarlike functions has been extended from the unit disk to other disk and half planes containing the origin by Ruscheweyh and Salinas(see [2]). Let be one such disk or half plane.Then there are two unique parameters C \ {0} and [0, 1] such that
where,
, = {w,(z) : z } (1.3)
z w,(z) = 1 z .
Note that 1 / , iff  +  1.
Definition 1.1. (see[1][2][3]) Let 1, and = , for some admissible pair (, ). A function f H1(,) is called prestarlike of order in , if
1
f,(z) = f (w,(z)) R (1.4)
The set of all such functions f is denoted by R().
Let be the slit domain C \ [1, )(the slit being along the positive real axis).
Definition 1.2.(see[1][2][3]) Let 1. A function f H1() is called universally prestarlike of order if and only if f is prestarlike of order in all sets , with  +  1. The set of all such functions is denoted by Ru .
Note1.1.(see[2]) LetF (z) =
k =0
akzk =
1 dÂµ(t)
0
1 tz where ak =
1
tkdÂµ(t),
0
Âµ(t) is a probability measure on [0, 1]. Let T denote the set of all such
functions F . They are analytic in the slit domain .
Lemma 1.3.(see [6]) Let w(u, v) be a complex valued function, that is
w : D C (D C Ã— C)
and let u = u1 + iu2 and v = v1 + iv2
Suppose that the function w(u, v) satisfies the following conditions:

w(u, v) is continuous in D;
2. (1, 0) D and Re{w(1, 0)} > 0;
3. Re{w(iu2, v1)} 0 for all (iu2, v1) D and such that
21
(1 + u2)
v
2
Let
p(z) = 1 + p1z + p2z2 + . . .
be regular in such that
for all z . If then
(p(z), zp,(z)) D
Re{w(p(z), zp,(z))} > 0
Re{p(z)} > 0.
Some Properties of Universally prestarlike functions are discussed in (see[4][5]).
Theorem 2.1.If f H1() satisfies
> 1
( D+2f (z) 1
D+1f (z)
D+1f (z)
2
( 1
(z , = 2 2, 0 < 1.) for some 1 ( 1 1 < 1), then
D+1f (z)
D f (z) >
where,
(21( + 2) 3) + (21( + 2) 3)2 + 8( + 1)
j
=
4( + 1)
(2.0)
Hence f Ru . The result is Sharp.
P r o o f. It is known that for 0
z(Df (z)), = ( + 1)D+1f (z) Df (z) (2.1)
where (Df )(z) = z
(1z)
* f, for 0.In particular, for = n N. we
n!
have Dn+1f = z (zn1f )(n). This implies
z(Df (z)),
D f (z) = ( + 1)
If we define the function p(z) by
D+1f (z)
D+1f (z)
D f (z) (2.2)
D f (z) = + (1 )p(z) (2.3)
with defined as before (2.0), then
p(z) = 1 + p1z + p2z2 + . . .
is analytic in .
Now, differentiating both sides of equation (3.3) logarithmically, we have
D+2f (z)
z(Df (z),) (1 )p,(z)
( + 2) D+1f (z) = ( + 1) +
Now, using (2.1) in (2.4) we get,
D f (z) + + (1 )p(z) . (2.4)
D+2f (z) + 1 D+1f (z)
1 (1 )zp,(z)
D+1f (z) = + 2 which readily yields
D f (z) + + 2 + ( + 2)( + (1 )p(z))) (2.5)
Re
> 1.
( D+2f (z) 1
D+1f (z)
D+1f (z)
Therefore, if we define the function w(u, v) by
1
w(u, v) = ( +1) +( +1)(1)u(z)+1 ( +2)+ (1 )v(z)
(2.6)
then we see that
+ (1 )u(z)
1. w(u, v) is continuous in D = C \ [1, )
2. (1, 0) D and Re{w(1, 0)} = ( + 2)(1 1) > 0
3. for all (iu2, v1) D and such that
21
(1 + u2
v
2
Re{w(iu , v )} = ( + 1) + 1 ( + 2) + (1 )v1
2
2 1 1
2 + (1 )u2
2
2
(1 )(1 + u2)
( + 1) + 1 1( + 2) + 2(2 + (1 )u2)
Now, by simple computation and using (2.0) we get
2( + 1)2 (21( + 2) 3) 1 = 0
1
for 1 and 1 2 .
Hence Re{w(iu2, v1)} 0. This implies that the function w(u, v) satisfies
the hypothesis of lemma 1.3. Thus we conclude that
>
( D+1f (z) 1
D f (z)
D f (z)
which completes the proof.
Corollary 2.2. If = 2 2 0 and 0 1 < 1, 0 < 1, then
u u
R+1(1) R (( + 1)( ))
where, is defined as before in (2.0) and
( + 1)( ) 1
.
+1
P r o o f. Let f Ru
(1). Then we have
Re
> 1 (2.7)
( z(D+1f (z)), 1
D+1f (z)
D+1f (z)
By a simple computation, using (2.2) and (2.7), we obtain
( 1
D+2f (z)
Re D+1f (z)
> + 1 + 1 (2.8)
+ 2
Applying the theorem (2.1) we have
Re
> (2.9)
( D+1f (z) 1
D f (z)
D f (z)
where is defined as before in (2.0) Now, by a simple computation we get
z(Df (z)),
D f (z) =
( + 1)D+1f (z)
D f (z)
This implies
Hence
z(Df (z)),
D f (z) > ( + 1)( )
f Ru (( + 1)( ))
which completes the corollary.

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Department of Mathematics
Anna University Chennai,Chennai600025 India
email: shan@annauniv.edu
Department of Mathematics
Anna University Chennai,Chennai600025 India
email: lourthu mary@annauniv.edu