Internat. J. Math. & Math. Scf.
VOL. 14 NO. 3 (1991) 619-623
619
SOME SUFFICIENT CONDITIONS FOR UNIVALENCE
SUKHJIT SINGH
Department of Mathemat
Punjabl University
Patlala-147002 (Punjab)
India
(Received May 5, 1989 and in revised form
A new subclass R(u),
ABSTRACT.
starlike functions of order I/2
with
respect
to
the
Hadamard
0 < u < I, of
January I, 1990)
the
class
St(I/2)
the
is introduced and it is shown that
product
of
analytic
of
R(u) is closed
Some
functions.
class
sufficient
conditions for the normalized regular functions to be univalent in the unit disk E are
given.
KEY WORDS AND PHRASES.
Convex function, close-to-convex function, starlike function
of order I/2, univalent function, Hadamard product.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
30C45.
INTRODUCTION.
2
Let A denote the class of functions f(z)
z + a z +
which are regular in
2
the unit disk E
S
I}.
the
We
of A consisting of
denote
subclass
by
<
functions f which are univalent in E, K will stand for the usual subclass of S whose
members are convex in E. A function f e A is said to be close-to-convex in E if and
only if Re(f’(z)/g’(z)) > 0, z
Since g(z)
z is convex in E,
E, for some g e K.
the functions f e A which satisfy Re f’(z) > 0, z e E are close-to-convex in E. It is
{z/Iz
well known that every close-to-convex function in E is univalent in E.
u,
0 < u
<
1, denote by
St(a)
the subclass of
S consisting of
For a given
functions
f which
satisfy the condition
z
Re(Zf’f(z)
>
St(a)
c,, z c E.
is called the class of starlike functions of order u.
that for 0
<
a
< B
<
I,
St(B)
It is also well known
St(a).
In the present paper we introduce a new subclass R(u) of the class
St(I/2)
and
prove that R(u) is closed with respect to convolutlon/Hadamard product of analytic
functions.
Some sufficient conditions are given for a function f e A to be in the
class S.
S. SINGH
620
2.
PRELIMINARIES.
We shall need the following
g(z)
b z
n
are
functioffienoted
regular
E,
in
If f(z)
def[nltions and results.
then
their
a
convolution/Hadamard
n
and
z
product
is
the
by f * g and defined by the power series
(f * g)(z)-
a
nffiO
(2. t)
b z
n
n
Let a,b and c be any complex numbers with c neither zero nor a negative integer.
Then the hypergeometrlc function F(a,b;c;z) is defined In Ralnville [I, p. 45] by
(a) (b)
n
+
F(a,b;c;z)
JtCn nt
nffil
()n
where
n
zn
(2.2)
is the Pochhammer symbol defined by
(2.3)
(+l)...(,+n-l), if n : N
{1,2,3,...).
E.
It is known that the series on the right in (2.2) is convergent for z
.
Now we define the function O(a,c) by
(a,c;z)
(a)
n
g
n+l
(c
(2.4)
E).
0,-1,-2,...; z
n-’O
From (2.2) and (2.4) we immediately get
(a,c;z)
LEI,iI 2.1.
[I, p. 47].
F(a,b;c;z)
.
LEMMA 2.2.
fa(z)
(2.5)
zF(l,a;c;z)
If
Izl
<
r(c)
r(b) r(c-b)
and if Re(c)
>
Re(b)
f tb-l(l-t)c-b-l(1-tz)
>
-a
0,
(2.6)
dt
For a given real number a, let
n-az n,
z
n-I
LEMMA 2.3. Let f
(B).
member of
E.
Then f
St(l/2)
a
is convex whenever a ) 0.
and g
St(),
where 1/2
<
< 1.
Then f
*
g is a
St
LEMA 2.2 is due to Lewis [2] and Lemma 2.3 follows the Corollary
by taking a- 1/2.
LEMMA 2.4. If f K, then Re(f(z)/z) > 1/2, z E.
in Lewis
[3]
SOME SUFFICIENT CONDITIONS FOR UNIVALENCE
and Re p(z)
If p(z) is analytic in E, p(O)
LEMMA 2.5.
621
>
1/2, z
E, then for
any function F, analytic in E, the function P * F takes values in the convex hurl of
F(E).
Lemma 2.4. is due to Strohhcker [4] and the assertion of Lemma 2.5 readily
follows by using Herglotz’ representation for P(z).
THEOREMS AND THEIR PROOFS.
3.
For 0
I, let R(a) denote the class of functions f
a
e A which
satisfy
the
condition
[
n
a
z
n
l+a.
* f(z) :
St(---)
n=l
THEOREM 3.1.
of
St(l/2)
PROOF.
and f e R(I) if and only if f(z)
St(l/2)
Clearly R(O)
(3.1)
z e E.
(i)
If 0
8
a
<
z.
1, then R(8) c__ R(a).
(ii) R(y) is a subclass
)0.(R)
for every "f
z +
Let f(z)
z
a
n
e R(8) so that
n=2
n 13 z
[.
gCz)
n
,
f(z)
nffil
(3.2)
St((l+8)/2).
Now
y.
n
azn *
.
where k(z)
nBz n
[
f(z)
n=l
* f(z)) *
ne-Sz n
[
n=l
n=l
(3.3)
g(z) * k(z),
n-(8-a)
Z
n
n=l
Since
B-a
O, therefore by Lemma 2.2, k(z) e K
9_
St(l/2).
In view of Lemma 2.3, we
now get from (3.2) and (3.3) that
g * k
S
((1 + 8)/2)c__
t
St((l+a)/2),
(as a
Hence from (3.3) and (3.1) we conclude that f e R(a).
part
(1).
The proof of part
observation that R(0)
PROOF.
Since f e
h(z)
[
This completes the proof of
(ll) follows immediately from part (1) and from the
St(I/2).
If f and g both belong to R(a), then f
THEOREM 3.2.
S).
* g
also belongs to R(a).
R(a), therefore
naz n *
f(z)
St((l+a)/2).
e
n--1
(3.4)
Now
naz n *
(f * g)(z)
7.
naz
* f(z)) * g(z)
n=l
h(z) * g(z).
(3.5)
S. SINGH
622
Since g e R(a)
c_
therefore in view of Lemma 2.3, we get, from (3.4) and (3.5)
St(I/2)
that
* g
h
St((l+a)/2),
E
which in turn Implies that
R(a).
* g
f
This completes the proof of our theorem.
y.
z +
If f(z)
THEOREM 3.3.
a
nffi2
+
Re
n=2
>
then Re f’(z)
0, z
E
A and satisfies the condition
z
n
>
n a z
n
(3.6)
I, z e E,
0, a
Hence f(z) is close-to-convex in E and therefore univalent
E.
in E.
We can wrlte
PROOF.
n-1
n=2
Now
by
Lemma
2.2,
n-I
a
n
n
n=2
the
function
ka(Z)
z
k (z)
az
Re
zn-I
[ n=2Y --l]
>
+
.
(z
n/na-1
(3.7)
a-I
convex
is
[(o(a,8;z) *
be satisfied.
I.
(3.8)
1/2.
n
Thus, from (3.6), (3.7), (3.8) and Lemma 2.5, we conclude that Re f’(z)
THEOREM 3.4. Let f e A and let for O
a, the condition
8
Re
for a
n=2
Therefore, in view of Lemma 2.4,
Re
+
n=2 n
f(z))
>
Then Re f’(z)
’]
>
1/2,
0, z e E.
>
0.
(3.9)
z e E,
Hence f(z) is close-to-convex in E and
therefore unlvalent in E.
PROOF.
The case when a
B
a.
is obvious, therefore we let
We can write
(3.10)
Now from (2.5) and Lemma 2.1, we have
0(,a;z)
z
F(l,;a;z)
r(a)
r(s) r(a-s)
f0 t
(8-1)
(l-t) a--I (1-tz)
-I
dr.
623
SOME SUFFICIENT CONDITIONS FOR UNIVALENCE
Since Re
[t B-I
l-t
=-8-1 l-t z
-I]
>
0 for all t, 0
<
t
<
and for all z, z c E, it
follows that
(8a;z)]
Re
>
(3.11)
O, z e E.
Form (3.9), (3.10), (3.11) and Lemma 2.5 the assertion of the theorem now follows.
ACKNOWLEDGEMENTS.
and comments.
The author wishes to thank the referee for his valuable suggestions
He is also grateful to Professor Ram Singh for his help and guidance.
REFERENCES
I.
RAINVILLE, E.D., Special Functions, Chelsea Publishing Company, Bronx, New York,
2.
LEWIS, J.L., Convexity of Certain Series, J. London Math. Soc. 27(1983), 435-446.
1960.
LEWIS, J.L., Convolutions of Starlike Functions, Indiana Univ. Math. Jour. 27(4)
(1978), 671-688.
4.
STROHHACKER, E.,
Beltrage
(1933), 356-380.
zur
Theorle dee
schl[chten
Funktlonen,
Math.
Z. 37