Convolutions with Hypergeometric Functions

BULLETIN ^{of the} Bull. Malaysian Math. Sc. Soc. (Second Series)23 (2000) 153-161 MALAYSIAN

MATHEMATICAL SCIENCES SOCIETY

Convolutions with Hypergeometric Functions

M. ANBU DURAI AND R. PARVATHAM

The Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai – 600 005, India

Abstract. In this paper we study the behaviour of where

is the Gaussian Hypergeometric function and the * is usual Hadamard product. In the main result, we find conditions on and E so that belong to whenever .

) ( ) , , , ( ) ( )]

(

[^{Ia,b,c f z zF abc z}

*

) , , , (abc z F

B A c b

a, , , , [I_a,b,c(f)](z) S^*[A,B] 1

, ) ( )

(z R E E f

1. Introduction

Let Adenote the family of functions that are analytic in the interior of unit disk

¦

2

) (

n n nz a z z f

} 1 :

{ 

' z C z . Let gbe analytic and univalent in ' and f be analytic in ' then f(z) is said to be subordinate to g(z), written f % g if f(0) g(0) and

).

( ) (' g ' f

For 1dBAd1, let

¿¾

½

¯®

­ '

c

 z

Bz Az z

f z f f z

B A

S ,

1 1 ) (

) ] (

,

*[ A %

For A 1,B 1 we get the well known family S^* of starlike functions. We further get S[12J,1] S^*(J) and S^*(O,0) S_O^*. For E 1,define

>

@

Re 2 / 2, {

)

( ¸ ! '

¹

¨ ·

©

§



 e f z z

f

R S S ⁱ E

T

E A ^T .

Note that when E t0, we have R(E)S, the class of univalent functions in A . For each E0,R(E) contains also nonunivalent functions.

For any complex number ‘a’ we define the ascending factorial notation for and

) 1 ( ) 1 ( ) ,

(a n a a an nt1 (a,0) 1 for az0. The triangle inequality for is When ‘a’ is neither zero nor a negative integer, we can write

) ,

(a n |(a,n)| d (|a|, n)).

).

( / ) ( ) ,

(a n *na *a

The Gaussian hypergeometric function is defined as

C c b a n z

n c

n b n z a

c b a

F ⁿ

n

¦

, , , (

0

wherec is neither zero nor a negative integer. The following well known formula 0

) (

Re ), ( ) (

) ( ) ) (

1 , , ,

( !

*

*

*

* c a b

b c a c

c b a c c

b a

F (1.1)

will be used frequently. Univalence, starlikeness and convexity properties of have been studied in [6] and [8].

) , , , (a b c z zF

For fA , we consider the Hohlov convolution operator [2] I_a,b,c (f) given by )

( ) , , , ( ) ( ] ) (

[^{Ia,b,c f z zF a b c z} * ^{f z} where* stands for the usual Hadamard product of power series.

For Rec!Reb!0, it is known that

³

*

*

* ¹

0

1 1

) 1 ( ) 1 ) (

( ) (

) ) (

, , ,

( a

b c b

tz t dt

b t c b z c c b a

F .

We can write

³

*

*

* ¹

0

1 1

,

, (1 )

) ) (

1 ) (

( ) (

) ) (

( )]

(

[ ^{abc b c b} * _z ^a

dt z t

tz t f

b t c b z c

f

I .

This operator reduces to Bernardi operator

³

1

0

1 ( )

) 1 ( )

(z t f tz dt

B_f J ^J

for a 1,b 1J and c 2J with Re J !1. For J 1 and 2, respectively we get Alexander transform and Libera transform. These three operators are all examples of the situation where c ab in I_a,b,c(f). Also we have

1 , ) ( ] ) ( [ ) ) (

1

( _z ¹ * ^{f z I1, 1,1 f z n !}

z

n n which is known as Ruscheweyh differential,

studied in [7]. It represents the case cab with a 1, b n1 and c 1. Some more special cases of the operator I_a,b,c(f)can be found in [10].

P.T. Mocanu [3] obtained the range for J so that the Bernardi operator whenever . As a natural extension, here we determine conditions on andE, the transform by the hypergeometric function on the class

S*

B_f )

0 ( R

f A,B,a,b,c

) , , , (a b c z

F R(E) so that

. ] , [ )

( ^*

,

, f S A B

I_abc 

2. Auxiliary lemmas

We shall state the following Lemmas [4] which may be used in proving the main theorems.

Lemma 2.1. Let a,b,c!0. Then (i) for c ! a b 1,

»¼

« º

¬

ª

*

*

*

*

¦

0 c a b

ab b

c a c

c b a c n

n c

n b n a n

n

(ii) for c ! a b 2,

¦

¼ º

«¬ ª

*

*

*

*

0 2

1 3 )

2 , 2 (

) 2 , ( ) 2 , 1 ( ) ( ) (

) ( ) (

) , 1 ( ) , (

) , ( ) , ( ) 1 (

n c a b

ab b

a c

b a b

c a c

c b a c n

n c

n b n a

n .

Lemma 2.2. Let a,b,c!0 and for az1,bz1,cz1with c!max{0,ab1},

¦

¼

« º

¬

ª

*

*

*

*

0

) 1 ) (

( ) (

) ( ) 1 ( ) 1 )(

1 (

1 )

1 , 1 )(

, (

) , )(

, (

n

b c c a c

c b a c b

a n

n c

n b n

a .

Lemma 2.3. Let a,b,c!0. For bz1 and c!1b,

¦

0

)) ( ) 1 ( ) ( 1 (

1 )

1 (

1 ) , (

) , (

n

b c b c

c n

n c

n

b \ \

) ( / ) ( )

(x x x

where\ *c * .

3. Main theorems

Now let us study the action of the hypergeometric function on the classes R(E)andS.

Theorem 3.1. Let a,bC\{0}, |a|z1, |b|z1, cz1 and c ! |a| |b|. For 1dB Ad1, assume that

°¿

°¾

½

°¯

°®

­

*

*

*

*

) 1 ( ) 1 (

) (

) 1 ) ( 1 ) ( ( ) (

) ( ) (

b a

b a c B A

b c a c

c b a c

) 1 ( ) 1 (

) 1 ( ) 1 ( ) 1 ( 2 1 1 )

(

¿¾

½

¯®

­

d a b

c B A

A E (3.1)

Then the operator I_a,b,c(f) maps R(E) into S^*[A,B].

Proof. Let a, bC\{0} and c ! |a||b|, |a| z 1, |b| z 1 and cz1. Let be a function in

¦

2

) (

n n nz a z

z

f R(E). Then, it is well-known that

an n

) 1 (

2 E

d .

Consider where( , , , ) ( )

¦

* ^{f z z n Bnzn}

z c b a

zF B₁ 1 and for nt1,

n

n a

n n c

n b n B a

) 1 , 1 ( ) 1 , (

) 1 , ( ) 1 , (

A special case of Theorem 3 in [1] gives a sufficient condition for fS^*[A,B] is that

¦

2

. )}

1 ( ) 1 ( {

n

n A B

a A B

n Then we have to show that

¦

2

. )}

1 ( ) 1 ( {

n

n A B

B A B

n T

We have

n n

n c

n b n A a

B n T

n

) 1 ( 2 ) 1 , 1 ( ) 1 , (

) 1 , ( ) 1 , } ( ) 1 ( ) 1 ( {

2

E

d

¦

1 1

1

) : 1 , 1 ( ) , (

) , ( ) , ) (

1 ) ( , 1 ( ) , (

) , ( ) , ) (

1 ( ) 1 (

2 T

n n c

n b n A a

n n c

n b n B a

n

n °¿

°¾

½

°¯

°®

­

^E

¦

¦

7

Using the formula (1.1) and Lemma 2.2. we observe that

°¯

°®

­

*

*

* *

(1 )

) ( ) (

) ( ) ) (

1 ( ) 1 (

1 2 B

b c a c

c b a B c

T E

°¿

°¾ ½

*

*

* *

(1 )

) 1 ( ) 1 (

) 1 ( ) 1 ( ) 1 ( ) 1 ( ) ( ) (

) (

) ( ) ) (

1

( A

b a

c A b

a b c a c

b a c c b a A c

°¯

°®

­

»»

¼ º

««

¬ ª

*

*

* *

) 1 ( ) 1 (

) (

) 1 ) ( 1 ) ( ( ) (

) ( ) ) (

1 (

2 a b

b a c B A

b c a c

c b a E c

°¿

°¾ ½

( )

) 1 ( ) 1 (

) 1 ( ) 1

( A B

b a

c

A .

Then under the hypothesis (3.1) of the theorem we get

B B A

T A

T

d

d 2(1 ) ) ) (

1 (

1 2

E E ,

thereby showing that fS^*[A,B].

Note. ForA O, B 0we get, as a special case, Theorem 2.1 of [4].

Theorem 3.2. Let bC\{0}, c!0, |b| z 1 and c!1|b|. For 1d BA d1, assume that

) ) (

1 ( ) 2 ) ( ) 1 ( 1 ( ) 1 1 ) ( 1 (

) 1 ( ) 1

( A B A B

b c b c

A c b

c c

B

d

¸

¹

¨ ·

©

§

\ E

\ (3.3)

where \(x) *c(x)/*(x). Then the operator I_1,b,c(f) maps R(E)into S^*[A,B]. Proof. Putting a 1 in (3.2) we get

°¿

°¾

½

°¯

°®

­

¦

¦

1 1

1 ( , )( 1)

) , ) (

1 ) ( , (

) , ) (

1 ( ) 1 ( 2

n n c n n

n A b

n c

n B b

T E .

Using (1.1) and Lemma 2.3. we get

°¿

°¾

½

°¯

°®

­ ¸

¹

¨ ·

©

§

( ( 1) ( )) ( )

1 ) 1 1 ) ( 1 (

) 1 ( ) 1 ) ( 1 (

1 2 c c b A B

b A c b

c c

T E B \ \ .

Thus under the hypothesis (3.3) of the theorem we get TdT₁d(AB), there by showing that the operator I_1,b,c(f) maps R(E) into S^*[A,B].

Note. For A O, B 0, we get as a special case, Theorem 2.2. of [4].

From the proof of Theorems 3.1 and 3.2, we observe that for A 1, B 0. We need not treat the case a 1 separately neither we need the aestrictions b z1 and cz1. In this case, we have the following result.

Corollary 3.3. Let a,bC\{0}and c ! |a| |b|. Assume that

) 1 ( 2 1 1 ) ( ) (

) ( ) (

E d

*

*

*

*

b c a c

c b a

c .

Then the operator I_a,b,c(f)maps R(E) into S^*[1,0].

Let S :[0,1]oR be a nonnegative function normalized so that

³

1

0

1 ) (t dt S

³

1

0

) , ) ( ( ) ( )]

(

[ dt f A

t tz t f z

f

V_S S .

Let ³

³

1

0

) ( )

( s

s ds

t S and assume that t3(t)o0whento0. It is shown in [9] that the class S^*[A,B], 1d BA d1 can be characterized interms of convolutions that

) 0

* ( ) ] (

,

[ ^{( , )}

* œ z

 z

z h z

z B f

A S

f ^AB

where

1

; ) 1 ( 1 )

( ₂

) ,

(

»¼ º

«¬ ª

x z

Bz A

x z A

z

h _AB . Choose ₂

) 1 ( ) (

) 1 ( ) ) (

(

t B A

t A B

t A

G

.

From tg'(t)g(t)1 2G(t), we get

t t B

A A t

B A

t B A t B

g log(1 )

) (

) 1 ( 2 )

1 ( ) (

) 1 ( ) ( ) 1 ( ) 2

(

An application of Theorem 2.1 in [5] gives the following result.

Theorem 3.4. LetE be given by

³

¼

« º

¬

ª

1

0

) 1 log(

) (

) 1 ( 2 )

1 ( ) (

) 1 ( ) ( ) 1 ( ) 2

1 ( dt

t t B

A A t

B A

t B A t B

E S

E .

Then,

'

 t

œ

 S A B L₃ e h e z z R

V_S ( (E)) ^*[ , ] ( ^iT _(A,B)( ^iT )) 0,

Where

³

¼ º

««

¬ ª

¸¹

¨ ·

© 3 §

' 3 

1

0

)2

1 ( ) (

) 1 ( ) ( ) Re ( ) ( inf )

( dt

t B A

t A B A tz

tz t h

h L

z .

Note. The operator I_1,b,c(f) corresponds to V_S(f) with S(t) S_b,c(t)

1 1(1 ) )

( ) (

)

(

*

*

* _{b c b} t b t

c b

c where

³

1

0

,_c(t)dt 1

Sb A O, B 0 and

1 , 2

1 B

A J were treated in [4] and [5] respectively.

Next we determine the condition on a,b,c and A,B when f(z)is in S instead of )

( )

(z R E

f  .

Theorem 3.5. Let a,bC\{0}, c ! 2 a b . Suppose that a, b and satisfy the condition that

1

1d d

B A

««

¬ ª

*

*

*

*

) 1

( ) 2

(

) 1 ( ) 1 ( ) 1 ( )

( ) (

) ( ) (

b a c b a c

b b a a B b

c a c

c b a c

) ( 2 ) 1 (

) 3 2

( A B A B

b a c B ab

A d

»»

¼ º

(3.4)

Then the operator I_a,b,c(f)maps S into S^*[A,B].

Proof. Let aC {0},c! 2 a b and 1d B A d 1. Let . Then we have that

S z a z

z

f( )

¦

where

¦

2

) ( ) , , ,

( *

n n nz B z

z f z c b a zF

n

n a

n n c

n b n B a

) 1 , 1 )(

1 , (

) 1 , )(

1 , (

.

It is enough to show that

¦

2

. }

) 1 ( ) 1 ( {

n

n A B

B A B

n T

We have

n n

n a n c

n b n A a B

n

T

¦

2 ( , 1)(1, 1)

) 1 , ( ) 1 , )} ( 1 ( ) 1 ( {

¦

d

2

2

) , 1 ( ) , (

) , ( ) , } ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( {

n c n n

n b n A a

n B n

¦

¦

1

2 1

2

) : , 1 ( ) , (

) , ( ) , ( ) 1 ) (

1 ) (

, 1 ( ) , (

) , ( ) , ( ) 1 ) (

1 (

n n

n T n c

n b n a A n

n n c

n b n a B n

From Lemma 2.1. we get

«

«

¬ ª

*

*

*

*

b a c

ab B b

a c

b a B b B

c a c

c b a T c

1 ) 1 ( 3 ) 2 , 2

(

) 2 , ( ) 2 , ( ) 1 ( ) 1 ) ( ( ) (

) ( ) (

2

) 1 ( ) 1 ( ) 1 ) ( 1

( ) 1

( A B A

b a c

ab

A

»»

¼ º

««

¬ ª

*

*

*

*

b a c

ab A b

a c b a c

b b a a B b

c a c

c b a c

1 )

1 ( ) 2

(

) 1 ( ) 1 ( ) 1 ( )

( ) (

) ( ) (

( ) ( )

) 1

(

) 1 ( 3 ) 1

( A B A B

b a c

ab B b

a c

ab

»»

¼ º

««

¬ ª

*

*

*

*

) 1

( ) 2

(

) 1 ( ) 1 ( ) 1 ( )

( ) (

) ( ) (

b a c b a c

b b a a B b

c a c

c b a c

) ( ) 1 (

) 3 2

( A B A B

b a c

ab B

A

»»

¼ º

.

Then, under the hypothesis (3.4) of the theorem we get T dT₂ dAB. Therefore the operator I_a,b,c(f)maps S into S^*[A,B].

Note. When A O,B 0, this reduces to Theorem 2.6. in [4].

References

1. O.P. Ahuja, Families of analytic functions related to Ruscheweyh derivatives and subordinate to convex functions, Yokohama Math. J. 41 (1993), 39-49.

2. Y.E. Hohlov, Convolution Operators preserving univalence functions, Pliska Stud. Math. Bulgar.

10(1989), 87-92.

3. P.T. Mocanu, Starlikeness of certain integral operators, Mathematica, (Cluj)36 (59),2 (1994), 179-184.

4. S. Ponnusamy and F. Ronning, Starlikeness properties for convolutions involving Hypergeometric Series, Ann. Univ. Mariae Curie-Sklodowska sectA52 No.1 (1998), 141-155.

5. S. Ponnusamy and F. Ronning, Duality for Hadamard products applied to certain integral transforms,Complex Variables32(1997), 263-287.

6. S. Ponnusamy and M. Vuorinen, Univalence and convexity properties for Gaussian Hypergeometric functions, Rocky Mountain J. Math. (To appear).

7. St. Ruscheweyh, New Criteria for Univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109-115.

8. St. Ruscheweyh, and V. Singh, On the order of starlikeness of Hypergeometric functions, J. Maths. Anal. Appl.113 (1986), 1-11.

9. T. Shail-Small and E.M. Silvia, Neighborhoods of analytic functions, J. Analyse. Math.52 (1989), 210-240.

10. H.M. Srivastava, Univalence and Starlike integral operators and certain associated families of linear operators, Proceedings of the Conference on Complex Analaysis (Z. Li, F. Ren, L. Yang and S. Zhang, Eds.), International Press Inc., 1994.

Keywords: hypergeometric functions, starlikeness, subordination, Hadamard product.

1991 Mathematics Subjects Classification: 30C45, 33C05.

* The work was carried out when the first author is under the Faculty Improvement programme of Univesity Grants Commission of IX plan.