By logarithmic differentiation of P(z) = zf'(z)/f(z), we have. 1 +. Zfâk). ZP'(Z). -. = p(z) + - f'(z). Pb). = P(Z). (2.3). Let us put z = ~2. Then from (2.1)-(2.3), we have.
Applied Mathematics Applied Mathematics Letters 16 (2003) 653-655
PERGAMON
www.elsevier.com/locate/aml
A Certain Connection between
Starlike and Convex Functions N. TAKAHASHI AND M. NUNOKAWA Department of Mathematics, Gnnma University Aramaki, Maebashi Gunma 371-8510, Japan norihiroQed.edu.gunma-u.ac.jp mmokawaQedu.gunma-u.ac.jp (Received July 2001; revised and accepted September 200.2)
Abstract-we define two certain classes of functions S*(a, p) and C(a, p) and obtain a certain connection between these classes. @ 2003 Elsevier Science Ltd. All rights reserved.
Keywords-starlike,
convex.
1.
INTRODUCTION
Let A be the class of functions of the form
f(z) =
2 +
2 anzn, ?a=2
which are analytic in the open unit disc U = {z E C : 1.~1< 1). A function f(z) be starlike if it satisfies Re zf’(z) > 0
f(z)
’
in A is said to
z E u.
We denote by S* the subclass of A consisting of all starlike functions in U. A function f(z) is said to be convex if it satisfies Re(l++)
>0,
in A
zciI.J.
We denote by C the subclass of A consisting of all convex functions in U. Nunokawa, Owa, Saitoh, Cho and Takahashi [l] obtained the following result. Let p(z) be analytic in U with p(O) = 1 and p(z) # 0. If there exist two points z1 E U and 252E U such that LEMMA.
-- TP = argp(zl) < argp(z) < argp(z2) = y 2
The authors sincerely appreciate the helpful comments made by the referee. 0893-9659/03/$ - see front matter @ 2003 Elsevier Science Ltd. All rights reserved. PII: SO893-9659(03)00062-4
Typeset by A,@-‘&$
654
for(Y
N. TAKAHASHIAND
0,
>
/? >
M. NUNOKAWA
0, and for 1.z < Izl/ = 1~21,then we have ZlP’(Zl) ~z--_2-~ Pbl)
Q + P 2
and ZzP’(Z2) -----=~----_
f3 + P 2
P(Z2)
’
where m >
1 - I4
-
1 + IUI
and
From the lemma, we define two classes of functions. Let S*(cu,,S) be the subclass of A which
satisfies -
TP -7j--
1 - Ial - 1 + /a(’ By logarithmic
differentiation 1
+
-Zf”k) = p(z) f’(z)
ZP'(Z) + Pb)
Let us put z = ~2. Then from (2.1)-(2.3), 1 + zzf”(Z2)
=
p(Q)
f’(z2)
1 1+ -. P(Z2) (
= t2(a+B)Dei
(a/2)
Q
put
g(z)
=
5 = d/(2 + Q + p)/(2
$-l-(a+P)P
(2.3)
we have
ZzP’@2)
P(Z2) >
(
1 + t2-(a+a)12,-z
( us
= P(Z)
-
= t2(a+B)/2,2(“/2)a
Let
we have
of P(z) = zf’(z)/f(z),
- QI- p). Therefore,
i
(
1 + m. o+p 4
+ .-l-(~+mP,
(a/2) (I
. -.-.m a + p 4
t21++m
l+
b2 t2
>>
+ t2-l-b+w
(
ei(n/2) (1-a) >
z > 0. Then g(z) takes the minimum we have
1. value at
ZZf”(Z2) f’(z2)
(1 - Iol) fl(o,D) sin (n/2) (1 - o) 1 + loI + (1 - lol) u(o, P) cos (7F/2) (1 - o) ’
=y+T& where
This contradicts the assumption of the theorem. For the case z = ~1, applying the same method arg
l+w)
(1 - lol) a(o,D)
