Applied Mathematics Letters
Applied Mathematics Letters 15 (2002) 63-69
PERGAMON
Close-To-Convexity, of Certain
www.elsevier.com/locate/aml
Starlikeness, and Convexity Analytic Functions s. OWA
Department of Mathematics, Kinki University Higashi-Osaka, Osaka 577-8502, Japan owaQmath.kindai.ac.jp M. NUNOKAWA Department of Mathematics, Gunma University Aramaki (Maebashi), Gunma 371-8510, Japan nunokawaQedu. gunma-u. ac jp H. SAITOH Department of Mathematics, Gunma National College of Technology Toriba (Maebashi), Gunma 371-8530, Japan saitohQnat.gunma-ct.ac.jp H. M. SRIVASTAVA Department of Mathematics and Statistics, University of Victoria Victoria, British Columbia V8W 3P4, Canada
[email protected] (Received
December
2000; accepted
January
2001)
Abstract-The main object of the present paper is to derive several sufficient conditions for close-to-convexity, starlikeness, and convexity of certain (normalized) analytic functions. Relevant connections of some of the results obtained in this paper with those in earlier works are also provided. @ 2001 Elsevier Science Ltd. All rights reserved. Keywords-Analytic Subordination principle,
functions, Starlike functions, Univalent functions.
1. INTRODUCTION Let
A
denot,e
the
class
of functions
f
functions,
Convex functions,
AND DEFINITIONS
normalized
f(z)
Close-to-convex
= z+
hy
2 a,, ZR,
(1.1)
7t=2
which
are
analytic
in the
open
unit
disk
U := ‘The present
investigation
tinder Grant-in-Aid neering
Research
at the spring
was supported,
for General Council
meeting
Scientific
of Canada
{z
in part,
IzI
a, (Z E 24; 0 2 cr < 1)
,
(1.3)
and
where.
> cy, (Z E u; 0 2 a: < 1; g E Kc) )
(1.4)
for convenience,
s* := s*(o),
K := K(O),
Next, with a view to recalling the principle the functions f and g be analytic in U. Then there exists a function
h, analytic
and
c := C(0).
(1.5)
of subordination between analytic functions, let we say that the function f is subordinate to g if
in U, with
h(0) = 0
Ih(
and
< 1,
(z E U) 1
(1.6)
such that f(z) We denote
this subordination
= 9 (h(z)) >
(1.7)
by f(z)
In particular,
(z E U) .
if the function
g is univalent
+ 9(z).
(1.8)
in U, the subordination
(1.8) is equivalent
to (cf. [l,
11. 1901) f(O) = 9(O)
and
f(U)
c s(U).
(1.9)
results involving univalence and starRecently, Singh and Singh [4] p roved several interesting likeness of functions f E A. In our attempt here to generalize these results of Singh and Singh [4], we are led naturally to several sufficient conditions for close-to-convexity, starlikeness, and convexity of functions f E A. The following lemma (popularly known as Jack’s lemma) will be required in our present investigation. LEMMA j~(z)I
1. (See [5,6].) Let the (nonconstant)
where c is a real number
2.
function W(Z) be analytic in U with w(0) = 0. If value on the circle Iz/ = T < 1 at a point zo E U, then
attains its maximum
and c 2 1.
SUFFICIENT
Our first result functions f E A. THEOREM
CONDITIONS (Theorem
1. Let the function
1 below)
FOR provides
CLOSE-TO-CONVEXITY
a sufficient
condition
for close-to-convexity
of
f E A satisfy the inequality
(2.1)
Certain Analytic
65
Functions
Then (z E u: 0 2 cy < 1) )
(2.2)
or equivalently,
(F >,
fEC PROOF.
We begin
by defining
f’(z) = Then,
clearly,
a function
1+
now that there exists a point
by applying
0 5 Q < 1).
(2.4)
Lemma
(i E U).
- 1 + w(z)’
(2.5)
zo E U such that
and
when
iw(z)i < 1,
IzI < lzoi.
(2.6)
1, we have (c 2 1; ‘w (~0) = e”; Q E IR) .
20 w’ (20) = cw (x0), Thus,
z E u;
zw’( 2)
QZW’(Z)
1 +ckw(z)
lw (zo)l = 1 Then,
(w(z) # -1;
’
in U with w(0) = 0. We also find from (2.4) that
w is analytic
1 + -= z.f”b) f’(z)
Suppose
(2.3)
w by
1+ &W(Z) 1+ w(z)
(0 5 cy < 1).
(2.7)
we find from (2.5) and (2.7) that ITi
( 1
Zof"
+
f'
(zo) (zo)
)=lf%(g+?o) ca(a+cosQ)
=1+
-- c 2
1+a2+2ckcos6J 1+3a (20 E u; 0 5 f2 < 1) , = 2(1 + ck) ’
x+1 -
1, we shall obtain
2
- 1) (2 - X)
+ 2 (1 + x* - 2x cos f3) ’
(zo E u; 1 < x < 3) !
which yields the inequality ZOfll(ZO)
>
5x - 1 2(x+1)’
fl(iO)
=
X+1
CJ% 1 + (
Since (3.6) obviously
)
contradicts
(
(20EU;1
