In this paper, a boundary version of Schwarz lemma is investigated. For the function holomorphic
f
(
z
) =
a
+
c
_{p}
z^{p}
+
c_{p}
+
_{1}
z
^{p+1}
+ ... defined in the unit disc satisfying 
f
(
z
) − 1 < 1, where 0 <
a
< 2, we estimate a module of angular derivative at the boundary point
b
,
f
(
b
) = 2, by taking into account their first nonzero two Maclaurin coefficients. The sharpness of these estimates is also proved.
1. INTRODUCTION
In recent years, a boundary version of Schwarz lemma was investigated in D. M. Burns and S. G. Krantz
[8]
, R. Osserman
[11]
, V. N. Dubinin
[3
,
4]
, M. Jeong
[6
,
7]
, H. P. Boas
[1]
, X. Tang, T. Liu and J. Lu
[10]
, D. Chelst
[2]
, X. Tang and T. Liu
[9]
and other’s studies.
The classical Schwarz lemma states that an holomorphic function
f
mapping the unit disc
D
= {
z
: 
z
 < 1} into itself, with
f
(0) = 0, satisfies the inequality 
f
(
z
) ≤ 
z
 for any point
z
∈
D
and 
f'
(0) ≤ 1. Equality in these inequalities (in the first one, for
z
≠ 0) occurs only if
f
(
z
) = λ
z
, λ = 1
[5
, p.329].
Let
f
(
z
) =
a
+
c
_{p}z
^{p}
+
c
_{p+1}
z
^{p+1}
+ ...,
c_{p}
≠ 0, be a holomorphic function in the unit disc
D
, and 
f
(
z
) − 1 < 1 for 
z
 < 1, where 0 <
a
< 2. Consider the functions
and
Then
h
(
z
) and
φ
(
z
) are holomorphic functions on
D
with 
h
(
z
) < 1 and 
φ
(
z
) < 1 for 
z
 < 1 and
φ
(0) = 0. Therefore, from the Schwarz lemma, we obtain
and
The equality in (1.1) for some nonzero
z
∈
D
or in (1.2) holds if and only if
where
θ
is a real number (see
[6]
).
It is an elementary consequence of Schwarz lemma that if
f
extends continuously to some boundary point
b
with 
b
 = 1, and if 
f
(
b
) = 1 and
f'
(
b
) exists, then 
f'
(
b
) ≥ 1, which is known as the Schwarz lemma on the boundary.
R. Osserman
[11]
considered the case that only one boundary fixed point of
f
is given and obtained a sharp estimate based on the values of the function. He has first showed that
and
under the assuumption
f
(0) = 0 where
f
is a holomorphic function mapping the unit disc into itself and
b
is a boundary point to which
f
extends continuously and 
f
(
b
) = 1. In addition, the equality in (1.3) holds if and only if
f
is of the form
where
θ
is a real number and
α
∈
D
satisfies arg
α
= arg
b
. Also, the equality in (1.4) holds if and only if
f
(
z
) =
ze
^{iθ}
, where
θ
is a real number.
If, in addition, the function
f
has an angular limit
f
(
b
) at
b
∈
∂D
, 
f
(
b
) = 1, then by the JuliaWolff lemma the angular derivative
f'
(
b
) exists and 1 ≤ 
f'
(
b
) ≤ ∞ (see
[14]
).
Inequality (1.4) and its generalizations have important applications in geometric theory of functions (see, e.g.,
[5]
,
[14]
). Therefore, the interest to such type results is not vanished recently (see, e.g.,
[1]
,
[3]
,
[4]
,
[6]
,
[7]
,
[9]
,
[10]
,
[11]
,
[13]
and references therein).
Vladimir N. Dubinin
[2]
has continued this line of research and has made a refinement on the boundary Schwarz lemma under the assumption that
f
(
z
) =
c_{p}z^{p}
+
c
_{p+1}
z
^{p+1}
+ ..., with a zero set {
a_{k}
}.
X. Tang, T. Liu and J. Lu
[10]
established a new type of the classical boundary Schwarz lemma for holomorphic selfmappings of the unit polydisk
D^{n}
in ℂ
^{n}
. They extended the classical Schwarz lemma at the boundary to high dimensions.
Also, M. Jeong
[6]
showed some inequalities at a boundary point for different form of holomorphic functions and found the condition for equality.
2. MAIN RESULTS
In this section, we can obtain more general results on the angular derivatives of holomorphic function on the unit disc at boundary by taking into account
c_{p}
,
c
_{p+1}
and zeros of
f
(
z
)−
a
if we know the first and the second coefficient in the expansion of the function
f
(
z
) =
a
+
c_{p}z^{p}
+
c
_{p+1}
z
^{p+1}
+ .... We obtain a sharp lower bound of 
f'
(
b
) at the point
b
, where 
b
 = 1.
Theorem 2.1.
Let
f
(
z
) =
a
+
c_{p}z^{p}
+
c
_{p+1}
z
^{p+1}
+...,
c_{p}
≠ 0
be a holomorphic function in the unit disc
D
,
and

f
(
z
) − 1 < 1
for

z
 < 1,
where
0 <
a
< 2.
Assume that, for some
b
∈
∂D
,
f
has an angular limit
f
(
b
)
at
b
,
f
(
b
) = 2.
Then
Moreover, the equality in
(1.5)
occurs for the function
Proof.
Consider the functions
φ
(
z
) and
B
(
z
) are holomorphic functions in
D
, and 
φ
(
z
) < 1, 
B
(
z
) < 1 for 
z
 < 1. By the maximum principle for each
z
∈
D
, we have
Therefore,
is holomorphic function in
D
and 
p
(
z
) ≤ 1 for 
z
 < 1. In particular, we have
and
Moreover, it can be seen that for
b
∈
∂D
The function
is holomorphic in the unit disc
D
, 
ϕ
(
z
) < 1,
ϕ
(0) = 0 and 
ϕ
(
b
) = 1 for
b
∈
∂D
. From (1.3), we obtain
and since
we obtain
Since
h
(0) =
f
(0) − 1 =
a
− 1 and
h
(
b
) = 1 for
b
∈
∂D
,
So, we take the inequality (1.5).
Now, we shall show that the inequality (1.5) is sharp. Let
Then
and
Also,
and
Since 
c_{p}
 =
a
(2 −
a
), (1.5) is satisfied with equality. ☐
If
f
(
z
)−
a
have no zeros different from
z
= 0 in Theorem 2.1, the inequality (1.5) can be further strengthened. This is given by the following Theorem.
Theorem 2.2.
Let
f
(
z
) =
a
+
c_{p}z^{p}
+
c
_{p+1}
z
^{p+1}
+...,
c_{p}
> 0
be a holomorphic function in the unit disc
D
,
f
(
z
) −
a
has no zeros in D except z
= 0
and

f
(
z
) − 1 < 1 for 
z
 < 1, where 0 <
a
< 2. Assume that, for some
b
∈
∂D
,
f has an angular limit
f
(
b
)
at
b
,
f
(
b
) = 2.
Then
and
In addition, the equality in
(1.7)
occurs for the function
and the equality in
(1.8)
occurs for the function
where
0 <
c_{p}
< 1
and
ln
.
Proof.
Let
c_{p}
> 0. Let
φ
(
z
) and
p
(
z
) be as in the proof of Theorem2.1. Having inequality (1.6) in mind , we denote by ln
p
(
z
) the holomorphic branch of the logarithm normed by the condition
The composite function
is holomorphic in the unit disc
D
, Φ(
z
) < 1 for 
z
 < 1, Φ(0) = 0 and Φ(
b
) = 1 for
b
∈
∂D
. From (1.3), we obtain
and
It can be seen that
and
Therefore, we obtain
Replacing arg
^{2}
p
(
b
)
by zero, we take
and we obtain (1.7) with an obvious equality case. Similary, function Φ(
z
) satisfies the assumptions of the Schwarz lemma, we obtain
and
Therefore, we have the inequality (1.8).
Now, we shall show that the inequality (1.8) is sharp. Let
where
Then
Under the simple calculations, we take
Therefore, we obtain
☐
If
f
(
z
) −
a
have zeros different from
z
= 0, taking into account these zeros, the inequality (1.5) can be strengthened in another way. This is given by the following Theorem.
Theorem 2.3.
Let
f
(
z
) =
a
+
c_{p}z^{p}
+
c
_{p+1}
z
^{p+1}
+...,
c_{p}
≠ 0
be a holomorphic function in the unit disc D
,
and

f
(
z
) − 1 < 1
for

z
 < 1,
where
0 <
a
< 2.
Assume that, for some b
∈
∂D
,
f has an angular limit f
(
b
)
at b
,
f
(
b
) = 2.
Let
a
_{1}
,
a
_{2}
, ...,
a_{n}
be zeros of the function
f
(
z
) −
a
in D that are different from zero. Then we have the inequality
In addition, the equality in
(1.9)
occurs for the function
where
a
_{1}
,
a
_{2}
, ...,
a_{n}
are positive real numbers.
Proof
. Let
φ
(
z
) be as in the proof of Theorem2.1 and
a
_{1}
,
a
_{2}
, ...,
a_{n}
be zeros of the function
f
(
z
) −
a
in
D
that are different from zero.
is a holomorphic function in
D
and 
B
_{0}
(
z
) < 1 for 
z
 < 1. By the maximum principle for each
z
∈
D
, we have
The function
is holomorphic in
D
and 
k
(
z
) ≤ 1 for 
z
 < 1. In particular, we have
and
Moreover, it can be seen that
Besides, with the simple calculations, we take
The auxiliary function
is holomorphic in the unit disc
D
,
,
and
for
b
∈
∂D
. From (1.3), we obtain
Since
and
we take
Thus, we take the inequality (1.9) with an obvious equality case. ☐
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