1. INTRODUCTION
The classical Schwarz lemma gives information about the behavior of a holomorphic function on the unit disc D = {z : |z| <1} at the origin, subject only to the relatively mild hypotheses that the function map the unit disc to the disc and the origin to the origin. This lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions defined on the unit disc. In its must basic form, the familiar Schwarz lemma says this ([5], p.329):
Let D be the unit disc in the complex plane ℂ. Let f : D → D be a holomorphic function with f(0) = 0. Under these circumstances |f(z)| ≤ |z| for all z ∈ D, and |f′(0)| ≤ 1. In addition, if the equality |f(z)| = |z| holds for any z ≠ 0, or |f′(0)| = 1 then f is a rotation, that is, f(z) = zeiθ, θ real.
For historical background about the Schwarz lemma and its applications on the boundary of the unit disc, we refer to (see [1], [19]).
Let f(z) be holomorphic function in D, f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1.
Let
and
F(z) is holomorphic and ℜf(z) > 0 for |z| < 1 and hence
is holomorphic, |ϕ(z)| < 1 for |z| < 1 and ϕ(0) = 0. Thus, by Schwarz lemma, we obtain
The inequality in (1:1) is sharp with equality for the function
where −1 < a ≤ 0 and b is any integer ≥ 1.
It is an elementary consequence of Schwarz lemma that if f extends continuously to some boundary point to with |c| = 1, and if |f(c)| = 1 and f'(c) exists, then |f'(c)| ≥ 1. This result of Schwarz lemma and its generalization are described as Schwarz lemma at the boundary in the literature. This improvement was obtained in [20] by Helmut Unkelbach, and rediscovered by R. Osserman in [15] 60 years later.
In the last 15 years, there have been tremendous studies on Schwarz lemma at the boundary (see, [1], [3], [4], [6], [7], [9], [10], [15], [16], [17], [19] and references therein). Some of them are about the below boundary of modulus of the functions derivation at the points (contact points) which satisfies |f(c)| = 1 condition of the boundary of the unit circle.
In [15], R. Osserman offered the following boundary refinement of the classical Schwarz lemma. It is very much in the spirit of the sort of result.
Lemma 1.1. Let f : D → D be holomorphic function with f(0) = 0. Assume that there is a c ∈ ∂D so that f extends continuously to c, |f(c)| = 1 and f'(c) exists.
Then
Inequality (1.2) is sharp, with equality possible for each value of |f'(0)|.
Corollary 1.2. Under the hypotheses Lemma 1.1, we have
and
|f'(c)| > 1 unless f(z) = zeiθ, θ real.
Moreover, if f(z) = apzp + ap+1zp+1...., then
The equality in (1.4) occurs for the function f(z) = zp (z + γ) = (1 + γz) ; 0 ≤ γ ≤ 1.
Lemma 1.3 (Julia-Wolff lemma). Let f be a holomorphic function in D, f(0) = 0 and f(D) ⊂ D. If, in addition, the function f has an angular limit f(b) at b ∈ ∂D, |f(b)| = 1, then the angular derivative f'(b) exists and 1 ≤ |f'(b)| ≤ ∞ (see [18]).
D. M. Burns and S. G. Krantz [8] and D. Chelst [2] studied the uniqueness part of the Schwarz lemma. In M. Mateljević’s papers, for more general results and related estimates, see also ([11], [12], [13] and [14]).
Also, M. Jeong [7] showed some inequalities at a boundary point for different form of holomorphic functions and found the condition for equality and in [6] a holomorphic self map defined on the closed unit disc with fixed points only on the boundary of the unit disc.
2. MAIN RESULTS
Theorem 2.1. Let f(z) be holomorphic function in D, f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1. Assume that, for some c ∈ ∂D, f has an angular limit f (c) at c, ℜf (c) = α. Then
The inequality (2.1) is sharp, with equality for the function
where −1 < a ≤ 0 and b is any integer ≥ 1.
Proof. Consider the function
Then ϕ(z) is holomorphic function in the unit disc D, ϕ(0) = 0 and |ϕ(z)| < 1. In addition, for ,
and since ℜf (c) = α, we take
From (1:3), we obtain
Since , we have
Thus, we get
Now, we shall that the inequality (2.1) is sharp. Let
Then, we take
Since −1 < a ≤ 0 and b is any integer ≥ 1, (2.1) is satisfied with equality. □
Theorem 2.2. Let f(z) be holomorphic function in D, f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1. Assume that, for some c ∈ ∂D, f has an angular limit f (c) at c, ℜf (c) = α. Then
The equality in (2.2) occurs for the function
where −1 < a ≤ 0, b is any integer ≥ 1 and is an arbitrary number from [0, 1] (see (1.1)).
Proof. Let ϕ(z) be the same as in the proof of Theorem 2.1. From (1.2), for , we obtain
Since
we have
Therefore, we take the inequality (2:2).
To show that the inequality (2.2) is sharp, take the holomorphic function
Then
Since −1 < a ≤ 0 and b is any integer ≥ 1, (2.2) is satisfied with equality. □
If f(z) = b + ap (z − a)p + ap+1 (z − a)p+1 + ... is a holomorphic function in D f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1, then
Theorem 2.3. Let f(z) = b + ap (z − a)p + ap+1 (z − a)p+1 + ... is a holomorphic function in D f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1. Assume that, for some c ∈ ∂D, f has an angular limit f (c) at c, ℜf (c) = α. Then
Equality in (2.4) occurs for the function
where −1 < a ≤ 0, b is any integer ≥ 1 and is an arbitrary number from [0, 1] (see (2.3)).
Proof. Using the inequality (1.4) for the function ϕ(z), for , we obtain
Also, since
we may write
Thus, we take the inequality (2.4).
The equality (2.4) is obtained for the function
as show simple calculations.
In Theorem 2.3, the inequality (2.4) is obtained by adding the term ap of f(z) function. In the following theorem, the inequality (2.4) is obtained by adding ap and ap+1 that are consecutive terms of f(z) function.
Theorem 2.4. Let f(z) = b + ap (z − a)p + ap+1 (z − a)p+1 + ... , p ≥ 1 is a holomorphic function in D f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1. Assume that, for some c ∈ ∂D, f has an angular limit f (c) at c, ℜf (c) = α. Then
In addition, the equality in (2.5) occurs for the function
where −1 < a ≤ 0 and b is any integer ≥ 1.
Proof. Let ϕ(z) be the same as in the proof of Theorem 2.1. Also, let
We know that, from the maximum principle for each z ∈ D, we have |ϕ(z)| ≤ |zp|. Thus, k(z) is holomorphic function in D and |k(z)| < 1 for |z| < 1. In particular, we have
and
Moreover, we can see that, for ,
where ℏ(z) = zp.
The composite function
satisfies the assumptions of the Schwarz lemma on the boundary, whence we obtain
Because
and
we take
Therefore, we have
Now, we shall that the inequality (2.5) is sharp.
Consider the function
Then
and
Since −1 < a ≤ 0 and b is any integer ≥ 1, is satisfied with equality.
Also, because , (2.5) is satisfied with equality. □
If f(z) − b has no zeros different from z = a in Theorem 2.4, the inequality (2.5) can be further strengthened. This is given by the following Theorem.
Theorem 2.5. Let f(z) = b + ap (z − a)p + ap+1 (z − a)p+1 + ... p ≥ 1, ap > 0 is a holomorphic function in D f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1. and f(z) − b has no zeros in D except z = a. Assume that, for some c ∈ ∂D, f has an angular limit f (c) at c, ℜf (c) = α. Then
The equality in (2.7) occurs for the function
where −1 < a ≤ 0 and b is any integer ≥ 1.
Proof. Let ap > 0. Having in the mind inequality (2.6), we denote by ln k(z) the holomorphic branch of the logarithm normed by the condition
The auxiliary function
satisfies the assumptions of the Schwarz lemma on the boundary and so, for , we obtain
Since
and replacing arg2 k(z0) by zero, we take
Thus, we obtain the inequality (2.7).
The equality in (2.7) is obtained for the function
by simple calculations. □
Theorem 2.6. Under the same assumptions as in Theorem 2.5, we have
The equality in (2.8) occurs for the function
where −1 < a ≤ 0 and b is any integer ≥ 1.
Proof. Using the inequality (1.3) for the function Φ(z), we obtain
Replacing arg2 k(z0) by zero, we take
Therefore, we obtain the inequality (2.8) with an obvious equality case. □
참고문헌
- H.P. Boas: Julius and Julia: Mastering the Art of the Schwarz lemma. Amer. Math. Monthly 117 (2010), 770-785. https://doi.org/10.4169/000298910X521643
- D. Chelst: A generalized Schwarz lemma at the boundary. Proc. Amer. Math. Soc. 129 (2001), 3275-3278. https://doi.org/10.1090/S0002-9939-01-06144-5
- V.N. Dubinin: The Schwarz inequality on the boundary for functions regular in the disc. J. Math. Sci. 122 (2004), 3623-3629. https://doi.org/10.1023/B:JOTH.0000035237.43977.39
- V.N. Dubinin: Bounded holomorphic functions covering no concentric circles. J. Math. Sci. 207 (2015), 825-831. https://doi.org/10.1007/s10958-015-2406-5
- G.M. Golusin: Geometric Theory of Functions of Complex Variable [in Russian]. 2nd edn., Moscow 1966.
- M. Jeong: The Schwarz lemma and its applications at a boundary point. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 21 (2014), 275-284.
- M. Jeong: The Schwarz lemma and boundary fixed points. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 18 (2011), 219-227.
- D.M. Burns & S.G. Krantz: Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J. Amer. Math. Soc. 7 (1994), 661-676. https://doi.org/10.1090/S0894-0347-1994-1242454-2
- X. Tang & T. Liu: The Schwarz lemma at the boundary of the egg domain Bp1,p2 in ℂn. Canad. Math. Bull. 58 (2015), 381-392. https://doi.org/10.4153/CMB-2014-067-7
- X. Tang, T. Liu & J. Lu: Schwarz lemma at the boundary of the unit polydisk in ℂn. Sci. China Math. 58 (2015), 1-14.
- M. Mateljević: The lower bound for the modulus of the derivatives and Jacobian of harmonic injective mappings. Filomat 29:2 (2015), 221-244. https://doi.org/10.2298/FIL1502221M
- M. Mateljević: Distortion of harmonic functions and harmonic quasiconformal quasi-isometry. Revue Roum. Math. Pures Appl. 51 (2006) 56, 711-722.
- M. Mateljević: Ahlfors-Schwarz lemma and curvature. Kragujevac J. Math. 25 (2003), 155-164.
- M. Mateljević: Note on rigidity of holomorphic mappings & Schwarz and Jack lemma (in preparation). ResearchGate.
- R. Osserman: A sharp Schwarz inequality on the boundary. Proc. Amer. Math. Soc. 128 (2000), 3513-3517. https://doi.org/10.1090/S0002-9939-00-05463-0
- T. Aliyev Azeroğlu & B.N. Örnek: A refined Schwarz inequality on the boundary. Complex Variables and Elliptic Equations 58 (2013), 571-577. https://doi.org/10.1080/17476933.2012.718338
- B.N. Örnek: Sharpened forms of the Schwarz lemma on the boundary. Bull. Korean Math. Soc. 50 (2013), 2053-2059. https://doi.org/10.4134/BKMS.2013.50.6.2053
- Ch. Pommerenke: Boundary behaviour of conformal maps. Springer-Verlag, Berlin, 1992.
- M. Elin, F. Jacobzon, M. Levenshtein & D. Shoikhet: The Schwarz lemma: Rigidity and Dynamics. Harmonic and Complex Analysis and its Applications. Springer International Publishing, (2014), 135-230.
- H. Unkelbach: Uber die Randverzerrung bei konformer Abbildung. Math. Z. 43 (1938), 739-742. https://doi.org/10.1007/BF01181115