Properties Of Universally Prestarlike Functions

Properties Of Universally Prestarlike Functions

Properties Of Universally Prestarlike Functions

1. 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 dµ(t)

0

1 tz where ak =

1

tkdµ(t),

0

µ(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 × C)

and let u = u1 + iu2 and v = v1 + iv2

Suppose that the function w(u, v) satisfies the following conditions:

1. 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.

1. S. Ruscheweyh, Some properties of prestarlike and universally prestarlike functions, Intern. Conf. Geometric Function Theory, J. Analysis., 15 (2007), 247-254.

2. S. Ruscheweyh,L. Salinas, Universally prestarlike functions as convolu- tion multipiers, Mathematische Zeitschrift, (2009), 607-617.

3. S. Ruscheweyh, L. Salinas, T. Sugawa, Completely monotone sequences and universally prestarlike functions, Israel J. Math.,171 (2009), 285- 304.

4. T.N. Shanmugam, J. Lourthu Mary, Fekete-Szego inequality for univer- sally prestarlike functions, Fract. Calc. Appl. Anal.,13, No.4, 2010), 385-394.

5. Tirunelveli Nellaiappan Shanmugam, Joseph Lourthu Mary, A Note on universally prestarlike functions, Stud. Univ. Babes-Bolyai Math.,57, No. 1, (2012), 5360

6. S.S. Miller, P.T. Mocanu, Second order differential inequalities in the complex plane, J.Math.Anal.Appl.,65 (1978),289-305.

Department of Mathematics

Anna University Chennai,Chennai-600025 India

e-mail: shan@annauniv.edu

Department of Mathematics

Anna University Chennai,Chennai-600025 India

e-mail: lourthu mary@annauniv.edu