On the Growth of Analytic Functions in the Class U(λ)

• Published in: Computational methods and function theory Vol. 13; no. 4; pp. 613 - 634
• Main Authors: Vasudevarao, A., Yanagihara, H.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-12-2013
• Subjects: Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; 30C45; 30C50; Conformal center; Univalent function; Coefficient estimate; 30C80; Starlike function; 30C25; Region of variability; Schwarz’s lemma
• ISSN: 1617-9447; 2195-3724
• DOI: 10.1007/s40315-013-0045-8

For
0
<
λ
≤
1
, let
U
(
λ
)
be the class of analytic functions in the unit disk
D
with
f
(
0
)
=
f
′
(
0
)
-
1
=
0
satisfying
|
f
′
(
z
)
(
z
/
f
(
z
)
)
2
-
1
|
<
λ
in
D
. Then, it is known that every
f
∈
U
(
λ
)
is univalent in
D
. Let
U
~
(
λ
)
=
{
f
∈
U
(
λ
)
:
f
′
′
(
0
)
=
0
}
. The sharp distortion and growth estimates for the subclass
U
~
(
λ
)
were known and many other properties are exclusively studied in Fourier and Ponnusamy (Complex Var. Elliptic Equ.
52
(1):1–8,
2007
), Obradović and Ponnusamy ( Complex Variables Theory Appl.
44
:173–191,
2001
) and Obradović and Ponnusamy (J. Math. Anal. Appl.
336
:758–767,
2007
). In contrast to the subclass
U
~
(
λ
)
, the full class
U
(
λ
)
has been less well studied. The sharp distortion and growth estimates for the full class
U
(
λ
)
are still unknown. In the present article, we shall prove the sharp estimate
|
f
′
′
(
0
)
|
≤
2
(
1
+
λ
)
for the full class
U
(
λ
)
. Furthermore, we shall determine the region of variability
{
f
(
z
0
)
:
f
∈
U
(
λ
)
}
for any fixed
z
0
∈
D
\
{
0
}
. This leads to the sharp growth theorem, i.e., the sharp lower and upper estimates for
|
f
(
z
0
)
|
with
f
∈
U
(
λ
)
. As an application we shall also give the sharp covering theorems.