The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc
In this paper, a boundary version of the Carathéodory inequality is studied. For the function f(z), defined in the unit disc with f(0) = 0, R f(z) ≤ A, we estimate a modulus of angular derivative at the boundary point z0, Rf(z0) = A, by taking into account the first two nonzero Maclaurin coefficient...
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irk-123456789-1405562018-07-11T01:23:12Z The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc Örnek, B.N. In this paper, a boundary version of the Carathéodory inequality is studied. For the function f(z), defined in the unit disc with f(0) = 0, R f(z) ≤ A, we estimate a modulus of angular derivative at the boundary point z0, Rf(z0) = A, by taking into account the first two nonzero Maclaurin coefficients. The sharpness of these estimates is also proved. 2016 Article The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc / B.N. Örnek // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 4. — С. 287-301. — Бібліогр.: 9 назв. — англ. 1812-9471 DOI : doi.org/10.15407/mag12.04.287 Mathematics Subject Classification 2000: 30C80 http://dspace.nbuv.gov.ua/handle/123456789/140556 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In this paper, a boundary version of the Carathéodory inequality is studied. For the function f(z), defined in the unit disc with f(0) = 0, R f(z) ≤ A, we estimate a modulus of angular derivative at the boundary point z0, Rf(z0) = A, by taking into account the first two nonzero Maclaurin coefficients. The sharpness of these estimates is also proved. |
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Örnek, B.N. |
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Örnek, B.N. The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc Журнал математической физики, анализа, геометрии |
| author_facet |
Örnek, B.N. |
| author_sort |
Örnek, B.N. |
| title |
The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc |
| title_short |
The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc |
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The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc |
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The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc |
| title_full_unstemmed |
The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc |
| title_sort |
carathéodory inequality on the boundary for holomorphic functions in the unit disc |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2016 |
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http://dspace.nbuv.gov.ua/handle/123456789/140556 |
| citation_txt |
The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc / B.N. Örnek // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 4. — С. 287-301. — Бібліогр.: 9 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
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AT ornekbn thecaratheodoryinequalityontheboundaryforholomorphicfunctionsintheunitdisc AT ornekbn caratheodoryinequalityontheboundaryforholomorphicfunctionsintheunitdisc |
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2025-07-10T10:43:40Z |
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2025-07-10T10:43:40Z |
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| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2016, vol. 12, No. 4, pp. 287–301
The Carathéodory Inequality on the Boundary for
Holomorphic Functions in the Unit Disc
B.N. Örnek
Department of Computer Engineering, Amasya University
Merkez–Amasya 05100, Turkey
E-mail: nafiornek@gmail.com
Received February 12, 2013, revised December 13, 2015
In this paper, a boundary version of the Carathéodory inequality is
studied. For the function f(z), defined in the unit disc with f(0) = 0,
<f(z) ≤ A, we estimate a modulus of angular derivative at the boundary
point z0, <f(z0) = A, by taking into account the first two nonzero Maclaurin
coefficients. The sharpness of these estimates is also proved.
Key words: Schwarz lemma at the boundary, Carathéodory inequality.
Mathematics Subject Classification 2010: 30C80.
1. Introduction
In recent years, a boundary version of Schwarz lemma was studied by D.
Burns and S.G. Krantz ([2]), R.Osserman ([7]), V. N. Dubinin ([4]), B. N. Örnek
([8]) and others. On the contrary there was published the book ([1]), where the
authors studied the sharp real-part theorems (in particular, the Carathéodory
inequalities), which are frequently used in the theory of entire functions and in
the analytic function theory.
The Carathéodory inequality states that if the function f is holomorphic in
the unit disc D = {z : |z| < 1} with f(0) = 0 and <f ≤ A in D, then the
inequality
|f(z)| ≤ 2Ar
1− r
, |z| = r (1.1)
holds for all z ∈ D, and moreover,
∣∣f ′(0)
∣∣ ≤ 2A. (1.2)
c© B.N. Örnek, 2016
B.N. Örnek
The equality is achieved in (1.1) (for some nonzero z ∈ D ) or in (1.2) if and
only if f is a function of the form
f(z) =
2Azeiθ
1 + zeiθ
,
where θ is a real number ([1], p. 3, 4).
On the other hand, if f(z) = cpz
p + cp+1z
p+1.... p ∈ N is a holomorphic
function in the unit disc D = {z : |z| < 1} and if <f 6 A for |z| < 1, it can be
seen that the Carathéodory inequality can be strengthened by standard methods:
|f(z)| ≤ 2A |z|p
1− |z|p ,
and
|cp| ≤ 2A.
The classical Schwarz lemma states that a holomorphic function f , mapping
the unit disc D into itself with f(0) = 0, satisfies the inequality |f(z)| ≤ |z| for
any point z ∈ D, and |f ′(0)| ≤ 1. The equality in these inequalities (in the first
one, for z 6= 0) occurs only if f(z) = zeiθ, θ is a real number (see [5]). From
the Schwarz lemma, it is known that if a holomorphic function f , mapping the
unit disc into itself with f(0) = 0, extends continuously to a boundary point z0
with |z0| = 1, |f(z0)| = 1, and f ′(z0) exists, then |f ′(z0)| ≥ 1. This result of the
Schwarz lemma and its generalization are described in literature as the Schwarz
lemma at the boundary.
In ([7]), R. Osserman offered the following boundary refinement of the classical
Schwarz lemma. It is very much in the spirit of the sort of result we wish to
consider here. That is, ∣∣f ′(z0)
∣∣ > 2
1 + |f ′(0)| (1.3)
and ∣∣f ′(z0)
∣∣ ≥ 1, (1.4)
under the assumption f(0) = 0, where f is a holomorphic function mapping the
unit disc into itself and z0 is a boundary point to which f extends continuously,
and |f(z0)| = 1. Moreover, the equality in (1.4) holds if and only if f(z) = zeiθ,
where θ is a real number. Also, z0 = 1 in inequality (1.3) , the equality occurs
for the function f(z) = z (z + a) / (1 + az) , 0 ≤ a ≤ 1.
If, in addition, the function f has an angular limit f(z0) at z0 ∈ ∂D, |f(z0)| =
1, then by Julia–Wolff lemma, the angular derivative f ′(z0) exists, and 1 ≤
|f ′(z0)| ≤ ∞ ([9]).
288 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
The Carathéodory Inequality on the Boundary for Holomorphic Functions
Moreover, if f(z) = cpz
p + cp+1z
p+1...., then
∣∣f ′(z0)
∣∣ > p +
1− |cp|
1 + |cp| . (1.5)
Moreover, the equality in (1.5) occurs for the function f(z) = zp (z + γ) / (1 + γz),
0 ≤ γ ≤ 1.
We studied the boundary Carathéodory inequalities as an analog of the bound-
ary Schwarz lemma. We estimated a modulus of angular derivative of the function
that satisfied the Carathéodory inequality by taking into account the first two
nonzero Maclaurin coefficients.
2. Main Results
We have the following results.
Theorem 1. Let f(z) = cpz
p + cp+1z
p+1 . . . , cp 6= 0, p ≥ 2, p ∈ N, be a
holomorphic function in the unit disc D and let <f 6 A for |z| < 1. Further,
assume that for some z0 ∈ ∂D, f has an angular limit f(z0) at z0, <f(z0) = A.
Then the angular derivative f ′(z0) exists, and
∣∣f ′(z0)
∣∣ > A
2
(
p +
2 (2A− |cp|)2
4A2 − |cp|2 + 2A |cp+1|
)
. (1.6)
Moreover, the equality in (1.6) occurs for the function
f(z) = 2A
zp
1 + zp
.
P r o o f. Consider the functions
w(z) =
f(z)
2A− f(z)
, B(z) = zp.
The functions w(z) and B(z) are holomorphic in D, and |w(z)| ≤ 1, |B(z)| < 1
for |z| < 1.
That is,
|2A− f(z)|2 = |f(z)− 2A|2 = |f(z)|2 − 2< (f(z)2A) + 4A2
= |f(z)|2 − 4A< (f(z)) + 4A2.
From the hypothesis, since <f(z) 6 A and 4A<f(z) 6 4A2, we consider
|2A− f(z)|2 ≥ |f(z)|2 − 4A<f(z) + 4A<f(z) = |f(z)|2 .
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 289
B.N. Örnek
Therefore, we obtain ∣∣∣∣
f(z)
2A− f(z)
∣∣∣∣ ≤ 1.
By the maximum principle, for each z ∈ D, we have
|w(z)| ≤ |B(z)| .
Therefore,
t(z) =
w(z)
B(z)
is a holomorphic function in D, and |t(z)| ≤ 1 for |z| ≤ 1. In particular, we have
|t(0)| = |cp|
2A
≤ 1 (1.7)
and ∣∣t′(0)
∣∣ =
|cp+1|
2A
.
If |t(0)| = 1, then by the maximum principle, we have w(z)
B(z) = eiθ and f(z) =
2Azpeiθ
1+zpeiθ , where θ is a real number. For the function f(z), (1.6) holds. Further we
may assume f(z) 6≡ 2Azpeiθ
1+zpeiθ , and thus |t(0)| < 1.
Moreover, since the expression z0w′(z0)
w(z0) is a real number greater than or equal
to 1 (see [7]) and <f(z0) = A yields |w(z0)| = 1, we get
z0w
′(z0)
w(z0)
=
∣∣∣∣
z0w
′(z0)
w(z0)
∣∣∣∣ =
∣∣w′(z0)
∣∣ .
Also, since |w(z)| ≤ |B(z)|, we take
1− |w(z)|
1− |z| ≥ 1− |B(z)|
1− |z| .
Because f(z) has an angular limit at z0, then w(z) has an angular limit at z0
and from the Julia–Wolff lemma the function w(z) has an angular derivative at
z0. Passing to the angular limit in the last inequality yields
∣∣w′(z0)
∣∣ ≥ ∣∣B′(z0)
∣∣ .
Therefore, we obtain
z0w
′(z0)
w(z0)
=
∣∣w′(z0)
∣∣ ≥ ∣∣B′(z0)
∣∣ =
z0B
′(z0)
B(z0)
.
290 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
The Carathéodory Inequality on the Boundary for Holomorphic Functions
The composite function
T (z) =
t(z)− t(0)
1− t(0)t(z)
is holomorphic in the unit disc D, |T (z)| < 1, T (0) = 0, and |T (z0)| = 1 for
z0 ∈ ∂D.
From (1.3), we obtain
2
1 + |T ′(0)| ≤ ∣∣T ′(z0)
∣∣ =
1− |t(0)|2∣∣∣1− t(0)t(z0)
∣∣∣
2
∣∣t′(z0)
∣∣
=
1− |t(0)|2∣∣∣1− t(0)t(z0)
∣∣∣
2
∣∣∣∣
w′(z0)
B(z0)
− w(z0)B′(z0)
B2(z0)
∣∣∣∣
=
1− |t(0)|2∣∣∣1− t(0)t(z0)
∣∣∣
2
∣∣∣∣
w(z0)
z0B(z0)
∣∣∣∣
∣∣∣∣
z0w
′(z0)
w(z0)
− z0B
′(z0)
B(z0)
∣∣∣∣
≤ 1 + |t(0)|
1− |t(0)|
{∣∣w′(z0)
∣∣− ∣∣B′(z0)
∣∣}
and
2
1 + |T ′(0)| ≤
1 + |t(0)|
1− |t(0)|
{∣∣w′(z0)
∣∣− ∣∣B′(z0)
∣∣} . (1.8)
It can be seen that
T ′(z) =
1− |t(0)|2(
1− t(0)t(z)
)2 t′(z),
T ′(0) =
1− |t(0)|2(
1− |t(0)|2
)2 t′(0)
=
t′(0)
1− |t(0)|2
and
∣∣T ′(0)
∣∣ =
|t′(0)|
1− |t(0)|2 =
|cp+1|
2A
1−
( |cp|
2A
)2 =
2A |cp+1|
4A2 − |cp|2
.
Since w(z) has an angular derivative at z0, the function f(z) has an angular
derivative at z0. Thus, we take
∣∣w′(z0)
∣∣ =
2A |f ′(z0)|
|2A− f(z0)|2
.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 291
B.N. Örnek
Also, we have |B′(z0)| = p for z0 ∈ ∂D. Let us substitute the values of |T ′(0)|,
|w′(z0)|, |B′(z0)| and |t(0)| into (1.8). Therefore, we obtain
2
1 + 2A|cp+1|
4A2−|cp|2
≤ 1 + |cp|
2A
1− |cp|
2A
{
2A |f ′(z0)|
|2A− f(z0)|2
− p
}
=
2A + |cp|
2A− |cp|
{
2A |f ′(z0)|
|2A− f(z0)|2
− p
}
,
2
(
4A2 − |cp|2
)
4A2 − |cp|2 + 2A |cp+1|
2A− |cp|
2A + |cp| ≤
2A |f ′(z0)|
|2A− f(z0)|2
− p
and
2
(
4A2 − |cp|2
)
4A2 − |cp|2 + 2A |cp+1|
2A− |cp|
2A + |cp| + p ≤ 2A |f ′(z0)|
|2A− f(z0)|2
.
Since |2A− f(z0)|2 ≥ (< (2A− f(z0)))
2 = A2, we get
2
(
4A2 − |cp|2
)
4A2 − |cp|2 + 2A |cp+1|
2A− |cp|
2A + |cp| + p ≤ 2A |f ′(z0)|
|2A− f(z0)|2
≤ 2 |f ′(z0)|
A
and (
2 (2A− |cp|)2
4A2 − |cp|2 + 2A |cp+1|
+ p
)
A
2
≤ ∣∣f ′(z0)
∣∣ .
So, we get inequality (1.6) .
Now we shall show that inequality (1.6) is sharp. Let
f(z) = 2A
zp
1 + zp
.
Then
f ′(z) = 2A
pzp−1
(1 + zp)2
and
f ′(1) =
pA
2
.
Since |cp| = 2A, (1.6) holds.
Theorem 2. Let f(z) = cpz
p + cp+1z
p+1 . . . , cp > 0, p ≥ 2, p ∈,N, be a
holomorphic function in the unit disc D and f(z) have no zeros in D except
z = 0, and let <f(z) 6 A for |z| < 1. Further, assume that for some z0 ∈ ∂D,
292 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
The Carathéodory Inequality on the Boundary for Holomorphic Functions
f has an angular limit f(z0) at z0, <f(z0) = A. Then the angular derivative
f ′(z0) exists and
∣∣f ′(z0)
∣∣ > A
2
p−
2 |cp|
(
ln |cp|
2A
)2
2 |cp| ln
( |cp|
2A
)
− |cp+1|
, (1.9)
where
|cp+1| ≤ 2
∣∣∣∣cp ln
( |cp|
2A
)∣∣∣∣ . (1.10)
In addition, the equality in (1.9) occurs for the function f(z) = 2A zp
1+zp , and the
equality in (1.10) occurs for the function
f(z) = 2A
zpe
1+z
1−z
ln( cp
2A)
1 + zpe
1+z
1−z
ln( cp
2A)
,
where 0 < cp < 1 and ln
( cp
2A
)
< 0.
P r o o f. Let cp > 0. Let w(z), t(z) and B(z) be as in the proof of Theorem
1. Having in mind inequality (1.7) , we denote by ln t(z) the holomorphic branch
of the logarithm normed by the condition
ln t(0) = ln
( cp
2A
)
< 0.
The function
b(z) =
ln t(z)− ln t(0)
ln t(z) + ln t(0)
is holomorphic in the unit disc D, |b(z)| < 1, b(0) = 0, and |b(z0)| = 1 for
z0 ∈ ∂D. Since f(z) has an angular limit at z0, w(z) has an angular limit at
z0 and from Julia-Wolff lemma the function w(z) has an angular derivative at
z0. Thus, since w(z) has an angular derivative at z0, the function f(z) has an
angular derivative at z0. From (1.3), we obtain
2
1 + |b′(0)| ≤ ∣∣b′(z0)
∣∣ =
|2 ln t(0)|
|ln t(z0) + ln t(0)|2
∣∣∣∣
t′(z0)
t(z0)
∣∣∣∣
=
|2 ln t(0)|
|ln t(z0) + ln t(0)|2
∣∣t′(z0)
∣∣
=
|2 ln t(0)|
|ln t(z0) + ln t(0)|2
∣∣∣∣
w′(z0)
B(z0)
− w(z0)B′(z0)
B2(z0)
∣∣∣∣
=
−2 ln t(0)
ln2 t(0) + arg2 t(z0)
{∣∣w′(z0)
∣∣− ∣∣B′(z0)
∣∣}
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 293
B.N. Örnek
and
2
1 + |b′(0)| ≤
−2 ln t(0)
ln2 t(0) + arg2 t(z0)
{∣∣w′(z0)
∣∣− ∣∣B′(z0)
∣∣} . (1.11)
It can be seen that
b′(z) =
2 ln t(0)
(ln t(z) + ln t(0))2
t′(z)
t(z)
and ∣∣b′(0)
∣∣ =
1
|2 ln t(0)|
∣∣∣∣
t′(0)
t(0)
∣∣∣∣ =
1
−2 ln |cp|
2A
|cp+1|
|cp| .
Let us substitute the values of |b′(0)|, |w′(z0)|, |B′(z0)| and ln t(0) into (1.11).
We obtain
2
1− |cp+1|
2|cp| ln
( |cp|
2A
)
≤ −2 ln t(0)
ln2 t(0) + arg2 t(z0)
{
2A |f ′(z0)|
|2A− f(z0)|2
− p
}
.
Replacing arg2 t(z0) by zero, we have
2
1− |cp+1|
2|cp| ln
( |cp|
2A
)
≤ −2
ln t(0)
{
2A |f ′(z0)|
|2A− f(z0)|2
− p
}
and
2 |cp| ln
( |cp|
2A
)
2 |cp| ln
( |cp|
2A
)
− |cp+1|
≤ −1
ln
( |cp|
2A
)
{
2A |f ′(z0)|
|2A− f(z0)|2
− p
}
.
Since |2A− f(z0)|2 ≥ (< (2A− f(z0)))
2 = A2, we get
p−
2 |cp|
(
ln |cp|
2A
)2
2 |cp| ln
( |cp|
2A
)
− |cp+1|
≤ 2A |f ′(z0)|
|2A− f(z0)|2
≤ 2 |f ′(z0)|
A
.
Thus, we obtain (1.9) with an obvious equality case.
Similarly, the function b(z) satisfies the assumptions of the Schwarz lemma
([5]), we obtain
1 ≥ ∣∣b′(0)
∣∣ =
|2 ln t(0)|
|ln t(0) + ln t(0)|2
∣∣∣∣
t′(0)
t(0)
∣∣∣∣
and
1 ≥ −1
2 ln
( |cp|
2A
) |cp+1|
|cp| .
294 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
The Carathéodory Inequality on the Boundary for Holomorphic Functions
Therefore, we have inequality (1.10).
Now we shall show that inequality (1.10) is sharp. Let
f(z) = zpg(z),
where
g(z) = 2A
e
1+z
1−z
ln( cp
2A)
1 + zpe
1+z
1−z
ln( cp
2A)
.
Then
g(0) = cp
and
g′(0) = cp+1.
After simple calculations, we get
cp+1 = 2cp ln
( cp
2A
)
.
Thus we obtain
|cp+1| = 2
∣∣∣∣cp ln
( |cp|
2A
)∣∣∣∣ .
We note that inequality (1.3) has been used in the proofs of Theorems 1 and
2. Therefore, there are both cp and cp+1 in the right-hand side of the inequalities.
But, if we use (1.4) instead of (1.3), we obtain a weaker but simpler inequality
(not including cp+1). It is formulated in the following theorem.
Theorem 3. Under the hypotheses of Theorem 2, we have
∣∣f ′(z0)
∣∣ > A
2
(
p− 1
2
ln
|cp|
2A
)
. (1.12)
The equality in (1.12) holds if and only if
f(z) = 2A
zpe
1+zeiθ
1−zeiθ ln( cp
2A)
1 + zpe
1+zeiθ
1−zeiθ ln( cp
2A)
,
where 0 < cp < 1, ln
( cp
2A
)
< 0, and θ is a real number.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 295
B.N. Örnek
P r o o f. From the proof of Theorem 2, using inequality (1.4) for the function
b(z), we obtain
1 ≤ ∣∣b′(z0)
∣∣ =
|2 ln t(0)|
|ln t(z0) + ln t(0)|2
∣∣∣∣
t′(z0)
t(z0)
∣∣∣∣
=
−2 ln t(0)
ln2 t(0) + arg2 t(z0)
{
2A |f ′(z0)|
|2A− f(z0)|2
− p
}
.
Replacing arg2 t(z0) by zero and since |2A− f(z0)|2 ≥ (< (2A− f(z0)))
2 = A2,
we get
1 ≤ ∣∣b′(z0)
∣∣ ≤ −2
ln
( |cp|
2A
)
(
2
A
∣∣f ′(z0)
∣∣− p
)
. (1.13)
Therefore, we have inequality (1.12).
If |f ′(z0)| = A
2
(
p− 1
2 ln |cp|
2A
)
, from (1.13) and |b′(z0)| = 1 we obtain
b(z) = zeiθ
and
ln t(z)− ln t(0)
ln t(z) + ln t(0)
= zeiθ.
Therefore, we take
ln t(z) =
1 + zeiθ
1− zeiθ
ln
( cp
2A
)
,
t(z) = e
1+zeiθ
1−zeiθ ln( cp
2A)
,
w(z)
B(z)
= e
1+zeiθ
1−zeiθ ln( cp
2A)
,
f(z)
2A− f(z)
= zpe
1+zeiθ
1−zeiθ ln( cp
2A)
and
f(z) = 2A
zpe
1+zeiθ
1−zeiθ ln( cp
2A)
1 + zpe
1+zeiθ
1−zeiθ ln( cp
2A)
.
Consider the product
Bn(z) =
n∏
k=1
z − ak
1− akz
.
296 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
The Carathéodory Inequality on the Boundary for Holomorphic Functions
The function Bn(z) is called a finite Blaschke product, where a1, a2, ..., an ∈
D. Let the function f(z) satisfy the conditions of the Carathéodory inequality and
also have zeros a1, a2, ..., an with order n1, n2, ....nk, respectively. Thus, one can
see that the Carathéodory inequality can be strengthened by standard methods
as follows:
|f(z)| ≤ 2A |z|p |Bn(z)|
1− |z|p |Bn(z)| (1.14)
and
|cp| ≤ 2A
n∏
k=1
|ak| . (1.15)
Inequalities (1.14) and (1.15) show that inequalities (1.1) and (1.2) can be strength-
ened if the zeros of the function which are different from the origin of f(z) in
inequality (1.6) are taken into account.
Theorem 4. Let f(z) = cpz
p + cp+1z
p+1 . . . , cp 6= 0, p ≥ 2, p ∈ N, be a
holomorphic function in the unit disc D and let <f(z) 6 A for |z| < 1. Assume
that for some z0 ∈ ∂D, f has an angular limit f(z0) at z0, <f(z0) = A. Let
a1, a2, . . . , an be the zeros of the function f(z) in D that are different from zero.
Then the angular derivative f ′(z0) exists, and
∣∣f ′(z0)
∣∣ > A
2
p +
n∑
k=1
1− |ak|2
|z0 − ak|2
+
2
(
2A
n∏
k=1
|ak| − |cp|
)2
4A2
(
n∏
k=1
|ak|
)2
− |cp|2 + 2A |cp+1|
n∏
k=1
|ak|
.
(1.16)
In addition, the equality in (1.16) occurs for the function
f(z) = 2A
1− 1
1 + zp
n∏
k=1
z−ak
1−akz
,
where a1, a2, . . . , an are positive real numbers.
P r o o f. Let w(z) be as in the proof of Theorem 1 and a1, a2, . . . , an be the
zeros of the function f(z) in D that are different from zero. The function
B1(z) = zp
n∏
k=1
z − ak
1− akz
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 297
B.N. Örnek
is holomorphic in D, and |B1(z)| < 1 for |z| < 1. By the maximum principle,
for each z ∈ D, we have
|w(z)| ≤ |B1(z)| .
The function
t1(z) =
w(z)
B1(z)
is holomorphic in D, and |t1(z)| ≤ 1 for |z| ≤ 1. In particular, we have
|t1(0)| = |cp|
2A
n∏
k=1
|ak|
≤ 1
and ∣∣t′1(0)
∣∣ =
|cp+1|
2A
n∏
k=1
|ak|
.
If |t1(0)| = 1, then, by the maximum principle, we have w(z)
B1(z) = eiϕ and
f(z) = 2A
zpeiϕ
n∏
k=1
z−ak
1−akz
1 + zpeiϕ
n∏
k=1
z−ak
1−akz
,
where ϕ is a real number. For the function f(z), (1.16) holds. Thus we may
assume
f(z) 6≡ 2A
zpeiϕ
n∏
k=1
z−ak
1−akz
1 + zpeiϕ
n∏
k=1
z−ak
1−akz
,
and |t1(0)| < 1.
It is obvious that
∣∣B′
1(z0)
∣∣ =
z0B
′
1(z0)
B1(z0)
= p +
n∑
k=1
1− |ak|2
|z0 − ak|2
.
Furthermore, since the expression z0w′(z0)
w(z0) is a real number greater than or equal
to 1 (see [7]) and <f(z0) = A yields |w(z0)| = 1, we get
z0w
′(z0)
w(z0)
=
∣∣∣∣
z0w
′(z0)
w(z0)
∣∣∣∣ =
∣∣w′(z0)
∣∣ .
298 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
The Carathéodory Inequality on the Boundary for Holomorphic Functions
Also, since |w(z)| ≤ |B1(z)|, we take
1− |w(z)|
1− |z| ≥ 1− |B1(z)|
1− |z| .
Since f(z) has an angular limit at z0, w(z) has an angular limit at z0 and from
Julia–Wolff lemma the function w(z) has an angular derivative at z0. Passing to
the angular limit in the last inequality yields
∣∣w′(z0)
∣∣ ≥ ∣∣B′
1(z0)
∣∣ .
Therefore, we obtain
z0w
′(z0)
w(z0)
=
∣∣w′(z0)
∣∣ ≥ ∣∣B′
1(z0)
∣∣ =
z0B
′
1(z0)
B1(z0)
.
The auxiliary function
T1(z) =
t1(z)− t1(0)
1− t1(0)t1(z)
is holomorphic in the unit disc D, |T1(z)| < 1, T1(0) = 0, and |T1(z0)| = 1 for
z0 ∈ ∂D. From (1.3), we obtain
2
1 + |T ′1(0)| ≤ ∣∣T ′1(z0)
∣∣ =
1− |t1(0)|2∣∣∣1− t1(0)t1(z0)
∣∣∣
2
∣∣t′1(z0)
∣∣
=
1− |t1(0)|2∣∣∣1− t1(0)t1(z0)
∣∣∣
2
∣∣∣∣
w′(z0)
B1(z0)
− w(z0)B′
1(z0)
B2
1(z0)
∣∣∣∣
≤ 1 + |t1(0)|
1− |t1(0)|
{∣∣w′(z0)
∣∣− ∣∣B′
1(z0)
∣∣}
and
2
1 + |T ′1(0)| ≤
1 + |t1(0)|
1− |t1(0)|
{∣∣w′(z0)
∣∣− ∣∣B′
1(z0)
∣∣} . (1.17)
It can be seen that
T ′1(z) =
1− |t1(0)|2(
1− t1(0)t1(z)
)2 t′1(z)
and
∣∣T ′1(0)
∣∣ =
|t′1(0)|
1− |t1(0)|2 =
|cp+1|
2A
n∏
k=1
|ak|
1−
|cp|
2A
n∏
k=1
|ak|
2 =
2A |cp+1|
n∏
k=1
|ak|
4A2
(
n∏
k=1
|ak|
)2
− |cp|2
.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 299
B.N. Örnek
Since w(z) has an angular derivative at z0, the function f(z) has an angular
derivative at z0. Let us substitute the values of |T ′1(0)|, |w′(z0)|, |B′
1(z0)| and
|t1(0)| into (1.17). Thus, we obtain
2
1 +
2A|cp+1|
n∏
k=1
|ak|
4A2
(
n∏
k=1
|ak|
)2
−|cp|2
≤
2A
n∏
k=1
|ak|+ |cp|
2A
n∏
k=1
|ak| − |cp|
{
2A |f ′(z0)|
|2A− f(z0)|2
− p−
n∑
k=1
1− |ak|2
|z0 − ak|2
}
,
2
[
4A2
(
n∏
k=1
|ak|
)2
−|cp|2
]
4A2
(
n∏
k=1
|ak|
)2
−|cp|2+2A|cp+1|
n∏
k=1
|ak|
2A
n∏
k=1
|ak|−|cp|
2A
n∏
k=1
|ak|+|cp|
+ p +
n∑
k=1
1−|ak|2
|z0−ak|2 ≤
2A|f ′(z0)|
|2A−f(z0)|2
and since |2A− f(z0)|2 ≥ (< (2A− f(z0)))
2 = A2, we get
2
(
2A
n∏
k=1
|ak| − |cp|
)2
4A2
(
n∏
k=1
|ak|
)2
− |cp|2 + 2A |cp+1|
n∏
k=1
|ak|
+ p +
n∑
k=1
1− |ak|2
|z0 − ak|2
≤ 2 |f ′(z0)|
A
.
Hence we get inequality (1.16).
Now we shall show that inequality (1.16) is sharp. Let
f(z) = 2A
1− 1
1 + zp
n∏
k=1
z−ak
1−akz
.
Then
f ′(z) = 2A
pzp−1
n∏
k=1
z−ak
1−akz +
n∑
k=1
1−|ak|2
(1−akz)2
n∏
s=1
k 6=s
z−as
1−asz zp
(
1 + zp
n∏
k=1
z−ak
1−akz
)2
and
f ′(1) = 2A
p
n∏
k=1
1−ak
1−ak
+
n∑
k=1
1−|ak|2
(1−ak)2
n∏
s=1
k 6=s
1−as
1−as
(
1 +
n∏
k=1
1−ak
1−ak
)2
.
300 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
The Carathéodory Inequality on the Boundary for Holomorphic Functions
Since a1, a2, . . . , an are positive real numbers, we have
f ′(1) =
2A
4
(
p +
n∑
k=1
1− a2
k
(1− ak)
2
)
=
A
2
(
p +
n∑
k=1
1 + ak
1− ak
)
.
Moreover, since |cp| = 2A
n∏
k=1
|ak| , (1.16) holds.
Acknowledgement. The author would like to thank the referees for their
constructive comments and suggestions on the earlier version of this paper.
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