On hadamard products for certain classes of univalent functions with negative coefficients

J. Korean Math. Soc.

Vol.19, No.2, 1983

ON HADAMARD PRODUCTS FOR CERTAIN CLASSES OF UNIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS

By SHIGEYOSHI OwA

I. Introduction

Let d denote the class of functions '"

f(z) =z+~ anz"

n=2

which are analytic and univalent in the unit disk 11= {z : IzI<I}. A function f(z) Ed is said to be starlike of order a (O~a<l), if

Re{z!'(z) }>a f(z)

for zE11 and a function f(z) Ed is said to be convex of order a (O~a<l),

if

Re{l +

for zE 11. And we denote by

d*

Let 75 denote the subclass of

d

DO

f(z) =z-~ anz" ^(an~O).

11=2

We also denote by 75* (a) and ~(a) the subclasses of 75 which are, res- pectively, starlike of order a _(O~a<l) in the unit disk 11 and convex of order a(O~a<l) in the unit disk 11.

For these classes, H. Silverman [6J showed the following lemmas.

LEMMA 1. A function

00

f(z) =z-~ a"z"

n=2

is in the class 75*(a) if and only if

'"

~ (n-a)all~l-a.

n=2

The result is sharp.

Received June 26, 1982.

76

LEMMA 2. A function

00

fez) =z-

1::

n=2

is in the class @(a) if and only if

00

1::n(n-a)an~l-a.

n=2

The result is sharp.

2. Properties of

and

THEOREM

I.

Proof.

Let a function

00

fez) =z-

1::

n=2

be in the class

and

Then, by using Lerruria 1, we have

00

1::(n-a2)an~1-a2

n=2

and

Hence we get

!;

f:

n=2 . 11=2 n=2 n=2

~1-a2+ e~1-a2)

-a2

This shows that

is in the class

by means of Lemma-I.

THEOREM

2.

The proof of Theorem 2 is obtained by using the. same technique as in the proof of Theorem 1 with the aid of Lemma 2.

3. Hadamard products

Let

denote the Hadamard product of two functions

00

fez) =z-

L:

n=2

and

00

g(z) =z-

1::

n=2

that is,

00

f*g(z) =z-1:;anbnzn.

n=2

THEOREM 3. Let the functions

00

fi(Z)=Z-1:;an,iZn (an.i;;;;O)

n=2

be in the same class @*(a) for every i=1,2, ..., m. Then the Hadamard product fl*f2*· ..*fm(z) belongs to the class @*(1- (1-a)mj (2-a)m-I).

Proof. Since fi(z)E@*(a) for every i=I,2,···m, by virtue of Lemma 1, we have

00

1:;(n-a)an,l~I-a

n=2

and

Further

0<1-

for O~a<1. This gives that the Hadamard product f₁*f2*,··*fm(z) is in the class @*(1-(1-a)mj(2-a)m-I).

COROLLARY 1. Under the hypotheses of Theorem

3,

Proof. Since

(l-a)m a<l- (2-a)m-1

for O~a<l, it is clear that fl*f2*···*fm(z) E@*(a) with the aid of The- orem 1.

THEOREM 4. Let the functions

00

fJz) =z-1:;an.ⁱZn

n=2

be in the same class @(a) for every i=l,2, ... , m. Then the Hadamard product fl*f2*···*fm(z) belongs to the class @(1- (l-a)m/2m-l (2-a)m-I).

(b ·:2::0)n.J-

Proof.

Since

for every

2, "',

conjunction with Lemma 2,

L:

71.=2

and

Again we have

. (l-a)m

0<1- 2

<1

for

This shows

is in the class

2

COROLLARY 2. Under the hypotheses of Theorem 4, we have fl*f2*· ..*fm (z) E@(a).

The proof of Corollary 2 is evident by using Theorem 2.

THEOREM

5.

, =

fi(Z) =z-:-

L:

n=2 .

be in the same class (6*(a) for every

i=l, 2,

=.

gj(z) =z-

L:

n=2

be in the same class @(a) for every j=l,

2, ...

(2-a)m+ p-l).

Proof.

Since

for every

and

'for every

2, ...

in virtue of Lemma 1 and Lemma 2, we have

i; (n-a)a1l.1~1-a,

n=2 n,.= 2-a

for

3, "',

and

b .< I-a

1I,J=o=2(2-a),

for

2 and

= 1, 2, "',

Hence we obtain

= { { (I-a)m+p }} ^{m p}

f2

^1-

DI

DI

= m p (l-a)m+p

~L;(n-a) .D_{0=2 .=1}^{a ,.,;}₎₌₁.Db,.,} ~ 2P(2-a)m+ p^{- l} _ _ { _ (I-a)m+ p }

-1 1 2P(2-a)m+ p^{- I}

and

(I-a)m+ p 0<1- 2P(2-a)m+ p ¹

<1

for

and

This completes the proof of the theorem.

COROLLARY 3. Under the hypotheses of Theorem 5, we have fl*f2*"'*fm*

-

gl*g2*"'*gp(z) E<O*(a).

THEOREM

6.

=

f;(z)=z-L; an,iZ" (an.i~O)

n=2

be in the same class <0* (a) for every i=1,2, . ", m. Further let the functions

=

gj(z) =z- L; bn,} z,. ^(bn.j~O)

n=2

be in the same class @(a) foreveryj=I,2,···,p. Then the Hadamard product fl*f2*···*fm*gl*g2*···*gp(z) belongs to the class @(l-(I-a)m+ p/2^P-1(2- a)m+p-l).

Proof.

Since

for every

and

for every

by using Lemma

and Lemma

a .< I-a

".'=

for

and

=

L; n(n-a)b,.,1;;;;1-a

n;:2

and

b .< I-a

n.J

=

for

and

3, "',

Consequently we have

~n{n-{I- 2P-~~;~~;::P-l lLVl

jDl

= m p (l-a)m+ p

;;;;L;n(n-a)_{0=2 .=1}

n

n

=1- 1- 2P 1(2-a)m+ p ^{1 •}

80

Further

(l-a)m+ p 0<1- 2P ¹(2-a)m+ p 1

<1

for

and

This shows that the Hadamard product

"'*fm*gl*g2*"'*gp(z)

is in the class

COROLLARY

4.

6,

gl*g2*"'gp(z) E@(a).

THEOREM

7.

co

fi(Z) =z-1; an, i Zn

11=2

(an,_i~O) be in the same class

75*

Proof.

Let

for every

by Lemma

we get

co

1; (n-a)an,l;:;;;l-a, (n-a)an,2;:;;;I-a

n=2

for

and

an,;;:;;;-2--I-a-a

for

and

Consequently we obtain

co { { (l-a)m}} '"

f2

¹

2

00 ' " (l-a)m

;:;;;1;(n-a)2Han,; < (2 )m-2

n=2 .=1 -a

{ (l-a)m}

-1- 1

- (2-a)m ²

and

(l-a)m 0<1- (2-a)m-2 <1

for

and

Hence we have the theorem with the aid of Lemma 2.

COROLLARY

5.

Theorem 7,

THEOREM

8.

00

f-(z) =z- "a_{a: £...J n,}t. zn

n=2

be in the same class @(a) for every i=l, 2, "', m. Then the Hadamard proauc~

f1*f2*'''*fm(z) belongs to the class

75*(1-

Proof.'

Since

for every

2, ".,

in virtue of Lemma 2, we get

and

a .< I-a n, , -:2-:-7(27""_-a):-

for

and

3, ".,

Therefore

~{{ (l-a)m }}'"

f2

I Dl

~ '" (I-a)m

~I;0=2(n-a) .=1

n

=1- {1- _2mg=~:-I}

and

(l-a)m 0<1- 2m(2-a)m I

<1

for

and

This gives that

is in the class

Cl - Cl

with Lemma

COROLLARY 6.

Theorem 8,

(a).

REMARK. For Hadamard products of other classes of analytic and univalent functions in the unit disk 11, S. Owa [3J, [4J showed some results.

4.

class

Let ,xl

be the class of function

F(z)=

~

+

where

The class ,xl (0) was studied by

N. Rahmanov [5J, A. E. Livingston [2J,

Gael [IJ and V. Singh and

Singh [7J.

In this place, let

be the class of function

G(z)

= _~

where

THEOREM 9.

G(z)

=-}

82

be in the class @l(a). Then G(z) Ew*(a).

Proof.

Let

fez) =z- L;a nzn= n=2

then the function

has the expansion

= (n+l) G(z) =z- L; - - anzn.

,,=2

2

Since

in conjunction with Lemma 2, we have

= (n+l)' =

'f2(n-a)

- 2 -

which shows that

belongs to the class

References

1. R.M. Gael, On a class of functions schlicht in the unit circle, Rev. Math. Hisp- Amer. 31 (1971), 20-33.

2. A. E. Livingston, On the radius of univalence of certain analytic functions, Proc.

Amer. Math. Soc. 17 (1966), 352-357.

3. S. Owa, A remark on the Hadamard products of starlike functions II, Math.

Japonica, 27 (1982), 747-752.

4. S. Owa, On the Hadamard products of univalent junctions, Tamkang

J.

14 (1983). (to appear).

5. B.N. Rahmanov, On the theory of univalent functions, DokI. Akad. Nauk. USSR 78 (1951), 209-211.

6. H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math.

Soc. 51 (1975), 109-116.

7. V. Singh and R. Singh, On a class of Junctions schlicht in the unit disc, Indian

J.

Department of Mathematics Kinki University

Osaka, Japan