SOOCHOW JOURNAL OF MATHEMATICS
Volume 25, No. 1, pp. 29-36, January 1999
MAPPING PROPERTIES OF ANALYTIC FUNCTION DEFINED
BY HYPERGEOMETRIC FUNCTION II
BY
NALINI SHUKLA AND PRATIBHA SHUKLA
Abstract. In the present paper, we determine the mapping properties of analytic
function h(z) in the unit disc U = fz : jzj < 1g, dened by
h(z) = (1 ; )(z) + z (z)
where 0 and (z) = zF (a b c z):
Our results generalize the recent results of Silverman 9] and Shukla and Shukla
8]. Further, we also found that our Theorem 3.4 at A = 2 ; 1 B = 1 and = 0
gives the corrected form of Theorem 4 due to Silverman.
0
1. Introduction
Let A denote the class of functions of the form
1 n
X
f (z ) = z +
n=2
anz
(1.1)
which are analytic in the unit disc U = fz : jz j < 1g.
Let ' ' (), K and K() denote respectively the class of starlike functions,
the class of starlike functions of order , the class of convex functions and the
class of convex functions of order in U , where 0 < 1: It is well known that
K() ' (), whenever 0 < 1.
Let H = fW : W is analytic, W (0) = 0, jW (z )j < 1 in Ug and let P (A B )
1+AW(z) ;1
denote the class o analytic function in U which are of the form 1+BW(z)
A < B 1, W 2 H: Dene
n
0
' (A B ) = f 2 A : zff 2 P (A B )g
Received June 17, 1996 revised December 6, 1997 revised September 24, 1998.
29
30
NALINI SHUKLA AND PRATIBHA SHUKLA
and
K(A B ) = ff 2 A : zf 0 2 ' (A B )g:
We observe that
' ((2 ; 1) ) '( )
' ( 1) ' ()
K((2 ; 1) ) K( )
K( 1) K()
' ((2 ; 1) 1) '()
' (;1 1) ' K((2 ; 1) 1) K()
and K(;1 1) K:
For a b and c are complex numbers with c is neither zero nor a negative
integer, let
1
X
(b)n z n (1.2)
F (a b c z) = ((ac))n(1)
n=0
n
n
denote the hypergeometric function, where ()n is the Pochhammer symbol dened by
(
( + 1) ( + n ; 1) n = 1 2 : : : ()n =
1
n = 0:
This function is analytic in the unit disc U . We also note that F (a b c 1) converges for Re(c ; a ; b) > 0 and is related to the gamma function by
c);(c ; a ; b) :
(1.3)
F (a b c 1) = ;(
;(c ; a);(c ; b)
Merkes and Scott 4] and Ruscheweyh and Singh 7] studied the mapping
properties of
(z) = zF (a b c z)
(1.4)
with the help of analytic theory of continued fraction. In 1987, Owa and Srivastava 6] studied the mapping properties of generalized hypergeometric function.
In 1990, Miller and Mocanu 5] determined the univalency of certain hypergeometric function by the method of dierential subordination. Recently, Silverman
9] and Shukla and Shukla 8] also investigated the mapping properties of (z )
with the help of elementary results of starlike and convex function. In fact, the
tremendous use of hypergeometric function in the recent proof of the Bieberbach
MAPPING PROPERTIES OF ANALYTIC FUNCTION
31
conjecture by Branges 2] in 1985 has prompted renewed interest in the properties
of this function.
In this paper, by the motivation of Bhoosnurmath and Swamy 1] and Branges
2], we study the mapping properties of the function h(z ) dened by
h(z) = (1 ; )(z) + z0(z)
(1.5)
where 0 and (z ) is dened by (1.4). In fact, in the investigation of mapping
properties of h(z ), our basic tools are the lemmas due to Goel and Sohi 3] at p = 1
which are given in Section 2. In Section 3, rstly, we nd the sucient condition
for h(z ) to be in J (A B ) and K(A B ). Further, we obtain the necessary and
sucient condition for h(z ) to be in J (A B ) and K(A B ) with appropriate
restrictions on a b c. We also investigate the mapping properties of the integral
operator of the form
Z z h(t)
I (z) =
(1.6)
t dt:
0
We will determine the sucient condition for I (z ) to be in K(A B ) and also
determine the necessary and sucient conditions for I (z ) to be in K(A B ) with
appropriate restrictions on a b c:
Our results generalize the corresponding results of Silverman 9] and Shukla
and Shukla 8]. Further, we also found that our Theorem 3.4 at A = 2 ; 1 B = 1
and = 0 gives the correct form of Theorem 4 of Silverman.
2. Preliminary Lemmas
We state Lemmas 2.1 and 2.2 due to Goel and Sohi 3] at p = 1 and prove
Lemma 2.3 that are needed in our investigation.
Lemma 2.1. A su cient condition for f (z) = z + P1n=2 anzn to be in
' (A B )(K(A B )) is that
1
X
f(1 + B )n ; (A + 1)gjan j
n=2
1
X
(B ; A)
nf(1 + B )n ; (A + 1)gjan j (B ; A) :
(2.1)
n=2
32
NALINI SHUKLA AND PRATIBHA SHUKLA
Lemma 2.2. Suppose f (z) = z ; P1n=2 janjzn. Then a necessary and su cient condition for f (z ) to be in ' (A B )(K(A B )), is that
1
X
f(1 + B )n ; (A + 1)gjan j
n=2
1
X
(B ; A)
nf(1 + B )n ; (A + 1)gjan j (B ; A) :
(2.2)
n=2
Lemma 2.3. If I (z) and h(z) dened by (1.6) and (1.5) respectively, then
I (z) 2 K(A B ) if and only if h(z) 2 ' (A B ).
Proof. We observe that
0 h
I 0 = hz I 00 = zhz;
2 and so
00
0
1 + z II 0 = zhh :
Thus any starlike result about h(z ) leads to a convex result about I (z ).
3. Main Results
The proof of each of the following theorems runs parallel to that of the
corresponding assertion made by Silverman 9] in the special case = 0 A =
2 ; 1 B = 1 and we omit the details involved.
Theorem 3.1. If a b > 0 and c > a + b + 2 then a su cient condition for
h(z) to be in ' (A B ) ;1 A < B 1, is that
;(c);(c ; a ; b) h1+f(1+ B )+ (1+2B ; A)gab+ (1 + B )(a)2 (b)2 i 2: (3.1)
;(c ; a);(c ; b)
(B ; A)(c ; a ; b ; 1) (B ; A)(c ; a ; b ; 2)2
h
Condition (3.1) is necessary and sucient for h1 dened by h1 (z ) = z 2; h(z)
z
to be in ' (A:B ):
i
Corollary (1). If we take A = (2 ; 1) B = and = 0, then Theorem
3:1 becomes:
MAPPING PROPERTIES OF ANALYTIC FUNCTION
33
If a b > 0 and c > a + b + 1 then a su cient condition for h(z ) to be in
' ( ), 0 < 1, 0 < 1 is that
i
;(c);(c ; a ; b) h1 +
(1 + )ab
;(c ; a);(c ; b)
2 (1 ; )(c ; a ; b ; 1) 2:
This condition is also necessary and su cient for h1 dened by
h1 (z) = z ;
to be in ' ( ).
1
X
n=2
(b)n;1 z n :
(1 ; + n) ((ac))n;1(1)
n;1
n;1
(3.2)
Remark. If we take A = ;1 B = 1 and = 0, the condition (3.1) is both
necessary and sucient for h1 to be in ' .
Theorem 3.2. If a b > ;1 c > 0 and ab < 0, then a necessary and su cient
condition for h(z ) to be in ' (A B ), is that
(1 + B )(a)2 (b)2 + f(1 + B ) + (1 + 2B ; A)gab(c ; a ; b ; 2)
+(B ; A)(c ; a ; b ; 2)2 0:
(3.3)
The condition c a + b + 1 ; ab is necessary and su cient for h(z ) to be in ' .
Corollary (2). If we take A = (2 ; 1) B = and = 0, then Theroem
3:2 becomes:
If a b > ;1 c > 0 and ab < 0, then a necessary and su cient condition for
h(z) to be in ' ( ), is that
c a + b + 1 ; (1 + )ab=2 (1 ; ):
The condition c a + b + 1 ; ab is necessary and su cient for h(z ) to be in '
Theorem 3.3. If a b > 0 and c > a + b + 3, then a su cient condition for
h(z) to be in K(A B ) ;1 A < B 1, is that
;(c);(c ; a ; b) h1 + f(2 + 3B ; A) + 2(1 + 2B ; A)gab
;(c ; a);(c ; b)
(B ; A)(c ; a ; b ; 1)
f
(1+
B
)
+
(4+5
B
;
A
)
g
(
a)2 (b)2 + (1 + B )(a)3 (b)3 i 2: (3.4)
+
(B ; A)
(c ; a ; b ; 2)2 (B ; A)(c ; a ; b ; 3)2
34
NALINI SHUKLA AND PRATIBHA SHUKLA
The condition (3:4) is necessary and su cient for h1 (z ) dened by (3:2) to be in
K(A B ).
Corollary (3). If we take A = (2 ; 1) B = and = 0, then Theorem
3:3 becomes:
If a b > 0 and c > a + b + 2, then a su cient condition for h(z ) to be in
K( ) 0 < 1 0 < 1, is that
;(c);(c ; a ; b) h1+ (2+4 ; 2 )
ab
(1 + )(a)2 (b)2 i 2:
+
;(c ; a);(c ; b)
2 (1 ; ) (c ; a ; b ; 1) 2 (1 ; )(c ; a ; b ; 2)2
This condition is also necessary and sucient for h1 dened by (3.2) to be
in K( ):
Theorem 3.4. If a b > ;1 ab < 0 and c > a + b + 3 then a necessary and
su cient condition for h(z ) to be in K(A B ), is that
(1+ B )(a)3 (b)3 + f(1 + B )+ (4+5B ; A)g(a)2 (b)2 (c ; a ; b ; 3)+ f(2+3B ; A)
+2(1+2B ; A)gab(c ; a ; b ; 3)2 +(B ; A)(c ; a ; b ; 3)3 0:
(3.5)
Corollary (4). If we take A = (2 ; 1) B = and = 0, then Theorem
3:4 becomes:
If a b > ;1 ab < 0 and c > a+b+2, then a necessary and su cient condition
for h(z ) to be in K( ), is that
(1 + )(a)2 (b)2 + (2 + 4 ; 2 )ab(c ; a ; b ; 2) + 2 (1 ; )(c ; a ; b ; 2)2 0:
Remark. The correct form of Theorem 4 of Silverman 9] from our Theorem
3.4 at A = 2 ; 1 B = 1 and = 0 is:
If a b > ;1 ab < 0 and c > a + b +2, then a necessary and sucient condition
for f (z ) to be in ' ()(K()), is that
(a)2 (b)2 + (3 ; )ab(c ; a ; b ; 2) + (1 ; )(c ; a ; b ; 2)2 0:
Theorem 3.5. If a b > 0 and c > a + b + 2 then a su cient condition for
I (z) dened by (1.6) to be in K(A B ) ;1 A < B 1 is that
;(c);(c ; a ; b) h1+f(1 + B )+ (1 + 2B ; A)g ab+ (1 + B )(a)2 (b)2 i 2:
;(c ; a);(c ; b)
(B ; A)(c ; a ; b ; 1)
(B ; A)(c ; a ; b ; 2)2
MAPPING PROPERTIES OF ANALYTIC FUNCTION
35
Corollary (5). If we take A = (2 ; 1) B = and = 0, then Theorem
3:5 becomes:
If a b > 0 and c > a + b + 1, then a su cient condition for I (z ) dened by
(1:6) to be in ' ( ) 0 < 1 0 < 1 is that
i
;(c);(c ; a ; b) h1 +
(1 + )ab
;(c ; a);(c ; b)
2 (1 ; )(c ; a ; b ; 1) 2:
Theorem 3.6. If a b > ;1 ab < 0 and c > a + b + 2, then a necessary and
su cient condition for I (z ) to be in ' (A B ), is that
(1 + B )(a)2 (b)2 + f(1 + B ) + (1 + 2B ; A)gab(c ; a ; b ; 2)
+(B ; A)(c ; a ; b ; 2)2 0:
Corollary (6). If we take A = (2 ; 1) B = and = 0 then Theorem
3:6 becomes:
If a b > ;1 ab < 0 and c > a + b +2 then a necessary and su cient condition
for h(z ) to be ' ( ), is that
c a + b + 1 ; ab(1 + )=2 (1 ; ):
Remarks.(i) If we take A = 2 ; 1 B = 1 and = 0, our results coincide
with the results of Silverman 9].
(ii) If we take = 0, our results coincide with the results of Shukla and
Shukla 8].
Acknowledgment
The authors are thankful to the referee, Dr. S. L. Shukla and Dr. A. M.
Chaudhary for their valuable suggestions.
References
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J. Pure Appl. Math., 12(1981), 738-742.
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36
NALINI SHUKLA AND PRATIBHA SHUKLA
3] R. M. Goel and N. S. Sohi, Multivalent functions with negative coe cients, Indian J. Pure
Appl. Math., 12(1981), 844-853.
4] E. P. Merkes and W. T. Scott, Starlike hypergeometric functions, Proc. Amer. Math. Soc.,
12(1961), 885-888.
5] S. S. Miller and P. T. Mocanu, Univalence of gaussian and conuent hypergeometric functions, Proc. Amer. Math. Soc., 110(1990), 333-342.
6] S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions,
Canad. J. Math., 39(1987), 1057-1077.
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Math. Anal. Appl., 113(1986), 1-11.
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Appl., 172(1993), 574-581.
Department of Mathematics, Janta College, Bakewar 206 124, Etawah (U.P.), INDIA.
Department of Mathematics, Christ Church College, Kanpur-208 001 (U.P.), INDIA.