In this paper, a boundary version of Carathéodory’s inequality is investigated. Also, new inequalities of the Carathéodory’s inequality at boundary are obtained and the sharpness of these inequalities is proved.
1. INTRODUCTION
In recent years, boundary version of Schwarz lemma was investigated in D. M. Burns and S. G. Krantz
[6]
, R. Osserman
[8]
, V. N. Dubinin
[2]
, M. Jeong
[4
,
5]
, H. P. Boas
[1]
and other’s studies. On the other hand, in the book
[7]
, Sharp RealParts Theorem’s (in particular Carathéodory’s inequalities), which are frequently used in the theory of entire functions and analytic function theory, have been studied.
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
) =
λz
, 
λ
 = 1 [3, p.329]. It is an elementary consequence of Schwarz lemma that if f extends continuously to some boundary point
z
_{0}
with 
z
_{0}
 = 1, and if 
f
(
z
_{0}
) = 1 and
f
'(
z
_{0}
) exists, then 
f
'(
z
_{0}
) ≥ 1, which is known as the Schwarz lemma on the boundary.
In this paper, we studied “boundary Carathéodory’s inequalities” as analog the Schwarz lemma at the boundary
[8]
.
The Carathéodory’s inequality states that, if the function
f
is holomorphic on the unit disc
D
with
f
(0) = 0 and ℜ
f
≤
A
in
D
, then the inequality
holds for all
z
∈
D
, and moreover
Equality is achieved in (1.1) (for some nonzero
z
∈
D
) or in (1.2) if and only if
f
(
z
) is the function of the form
where
θ
is a real number [7, pp.34].
Robert Osserman 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
z
_{0}
is a boundary point to which
f
extends continuously and 
f
(
z
_{0}
) = 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 argα = arg
z
_{0}
. Also, the equality in (1.4) holds if and only if
f
(
z
) =
ze^{iθ}
, where
θ
is a real number.
Moreover, if
then
It follows that
with equality only if
f
is of the form
f
(
z
) =
z^{p}e^{iθ}
,
θ
real
[8]
.
If, in addition, the function
f
has an angular limit
f
(
z
_{0}
) at
z
_{0}
∈
∂D
, 
f
(
z
_{0}
) = 1, then by the JuliaWolff lemma the angular derivative
f
'(
z
_{0}
) exists and 1 ≤ 
f
'(
z
_{0}
) ≤ ∞ (see
[11]
).
The inequality (1.5) is a particular case of a result due to Vladimir N. Dubinin in (see
[2]
), who strengthened the inequality 
f
'(
z
_{0}
) ≥ 1 by involving zeros of the function
f
. Some other types of strengthening inequalities are obtained in (see
[9]
,
[10]
).
We have following results, which can be offered as the boundary refinement of the Carathéodory’s inequality.
Theorem 1.1.
Let f be a holomorphic function in the unit disc D
,
f
(0) = 0
and
for

z
 < 1.
Further assume that
,
for some
z
_{0}
∈
∂D
,
f has an angular limit f
(
z
_{0}
)
at z
_{0}
,
ℜf
(
z
_{0}
) =
A
.
Then
Moreover
,
the equality in
(1.7)
holds if and if
wehere θ is a real number
.
Proof
. The function
is holomorphic in the unit disc
D
, 
ϕ
(
z
) < 1,
ϕ
(0) = 0 and 
ϕ
(
z
_{0}
) = 1 for
z
_{0}
∈
∂D
.
That is,

f
(
z
) − 2
A

^{2}
= 
f
(
z
)
^{2}
− 2ℜ (
f
(
z
)2
A
) + 4
A
^{2}
= 
f
(
z
)
^{2}
− 4
A
ℜ (
f
(
z
)) + 4
A
^{2}
.
From the hypothesis, since ℜ
f
(
z
) ≤
A
and 4
A
ℜ
f
(
z
) ≤ 4
A
^{2}
, we take
2
A
−
f
(
z
)2 ≥ 
f
(
z
)
^{2}
− 4
A
ℜ
f
(
z
) + 4
A
ℜ
f
(
z
) = 
f
(
z
)
^{2}
.
Therefore, we obtain
From (1.4), we obtain
So, we take
If
from (1.9) and 
ϕ
'(
z
_{0}
) = 1, we obtain
Theorem 1.2.
Let f be a holomorphic function in the unit disc D
,
f
(0) = 0
and
for

z
 < 1.
Further assume that
,
for some
z
_{0}
∈
∂D
,
f has an angular limit f
(
z
_{0}
) at
z
_{0}
, ℜ
f
(
z
_{0}
) =
A
.
Then
The inequality
(1.10)
is sharp
,
with equality for the function
where
is an arbitrary number on
[0, 1] (
see
(1.2)).
Proof
. Using the inequality (1.3) for the function (1.8), we obtain
and
Now, we shall show that the inequality (1.10) is sharp. Choose arbitrary
a
∈ [0, 1]
Let
Then
and
Since 
f
'(0) = 2A
a
, (1.10) is satisfied with equality.
An interesting special case of Theorem1.2 is when
f
''(0) = 0, in which case inequality (1.10) implies
Clearly equality holds for
θ
real.
Now, if
is a holomorphic function in the unit disc
D
and
for 
z
 < 1, it can be seen that Carathéodory’s inequality can be obtained with standard methods as follows:
and
The following result is a generalization of Theorem1.1.
Theorem 1.3. Let
c_{p}
≠ 0,
p
≥ 1
be a holomorphic function in the unit disc D and
for

z
 < 1.
Further assume that, for some z
_{0}
∈
∂D
,
f has an angular limit
f
(
z
_{0}
)
at z
_{0}
, ℜ
f
(
z
_{0}
) =
A
.
Then
In addition, the equality in
(1.12)
holds if and if
wehere θ is a real number
.
Proof
. Using the inequality (1.6) for the function (1.8), we obtain
Therefore, we take
If
from (1.13) and 
ϕ
' (
z
_{0}
) =
p
, we obtain
Theorem 1.4.
Under hypotheses of Theorem
1.3, we have
The inequality
(1.14)
is sharp
,
with equality for the function
where
is arbitrary number from
[0, 1] (
see
(1.11)).
Proof
. Using the inequality (1.5) for the function (1.8), we obtain
where
Since
we may write
Thus, we take
The equality in (1.14) is obtained for function
as show simple calculations.
Consider the following product:
B
(
z
) is called a finite Blaschke product, where
. Let the function
satisfy the conditions of Carathéodory’s inequality and also have zeros
a
_{1}
,
a
_{2}
, ...,
a
_{n}
with order
k
_{1}
,
k
_{2}
, . . . ,
k
_{n}
, respectively. Thus, one can see that Carathéodory’s inequality can be strengthened with the standard methods as follows:
and
The inequalities (1.15) and (1.16) show that the inequalities (1.1) and (1.2) will be able to be strengthened, if the zeros of function which are different from origin of
f
(
z
) in the (1.12) and (1.14) are taken into account.
Theorem 1.5.
Let
c_{p}
≠ 0,
p
≥ 1
be a holomorphic function in the unit disc D
,
and
for

z
 < 1.
Assume that for some z
_{0}
∈
∂D
,
f has an angular limit f
(
z
_{0}
)
at z
_{0}
, ℜ
f
(
z
_{0}
) =
A
.
Let a
_{1}
,
a
_{2}
, ...,
a_{n} be zeros of the
function f in D that are different from zero. Then we have the inequality
In addition, the equality in
(1.17)
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 Theorem1.1 and
a
_{1}
,
a
_{2}
, ...,
a_{n}
be zeros of the function
f
in
D
that are different from zero.
is a holomorphic functions in
D
, and 
B
(
z
) < 1
for

z
 < 1. By the maximum principle for each
z
∈
D
, we have

ϕ
(
z
) ≤ 
B
(
z
) .
The auxiliary function
is holomorphic in
D
, and 
φ
(
z
) < 1 for 
z
 < 1,
φ
(0) = 0 and 
φ
(
z
_{0}
) = 1 for
z
_{0}
∈
∂D
.
Moreover, it can be seen that
Besides, with the simple calculations, we take
From (1.5), we obtain
where
Since
we may write
Therefore, we have
Now, we shall show that the inequality (1.17) is sharp. Let
Then
and
Since
a
_{1}
,
a
_{2}
, ...,
a_{n}
are positive real numbers, we take
Since
(1.17) is satisfied with equality.
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