PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 101(115) (2017), 191–196 https://doi.org/10.2298/PIM1715191O

REPRESENTATION WITH MAJORANT OF THE SCHWARZ LEMMA AT THE BOUNDARY

Bülent Nafi Örnek and Tuğba Akyel

Abstract. Let f be a holomorphic function in the unit disc and |f(z)−1| < 1 for |z| < 1. We generalize the uniqueness portion of Schwarz’s lemma and provide sufficient conditions on the local behavior of f near a finite set of boundary points that needed for f to be a finite Blaschke product.

1. Introduction

Let f be a holomorphic function in the unit disc D = {z : |z| < 1}, f(0) = 0 and |f(z)| < 1 for |z| < 1. In accordance with the classical Schwarz lemma, for any point z in the disc D, we have |f(z)| ⩽ |z| and |f′(0)| ⩽ 1. Equality in these inequalities (in the first one, for z ≠ 0) occurs only if f(z) = λz, |λ| = 1 [6, p. 329].

In recent years, a boundary version of Schwarz lemma was investigated in Burns and Krantz [4], Chelts [5] and also in papers of a few other authors. They studied the uniqueness portion of the Schwarz lemma.

D. Burns and S. G. Krantz proved the following interesting theorem related to the uniqueness part of the classical Schwarz lemma for single variable functions [6].

Theorem 1.1. Let f : D → D be a holomorphic function such that

(1.1)  f(z) = z + O((z − 1)^{4})

as z → 1. Then f(z) = z on the disc.

Chelst [5] takes the Burns–Krantz theorem much further and takes the Blaschke product instead of the model function f(z) = z.

Theorem 1.2. Let f : D → D be a holomorphic function and ϕ : D → D a finite Blaschke product which equals τ ∈ ∂D on a finite set A_{f} ⊂ ∂D. If (i) for a given γ_{0} ∈ A_{f} , f(z) = ϕ(z)+O((z −1)^{4}), as z → γ_{0}, and (ii) for all γ ∈ A_{f} −{γ_{0}}, f(z) = ϕ(z) + O((z − 1)^{k_{γ}} ), for some k_{γ} ⩾ 2 as z → γ, then f(z) = ϕ(z) on the disc.

2010 Mathematics Subject Classification: Primary 30C80; Secondary 32A10.

Key words and phrases: Schwarz lemma, holomorphic function, Blaschke product, Hopf’s lemma on the disc.

Communicated by Miodrag Mateljević.

Let 𝔑 be a class of functions μ : (0, +∞) → (0, +∞) for each of which logμ(x) is concave with respect to logx. For each function μ ∈ 𝔑 the limit

μ_{0} = lim_{x→0} \frac{logμ(x)}{logx} ,

exists, and −∞ < μ_{0} ⩽ +∞. Here, the function μ ∈ 𝔑 is called bilogaritmic concave majorant [12].Obviously x^{α} ∈ 𝔑 for any α > 0.

In this paper, more general majorants will be taken instead of power majorants in conditions (i), (ii) and (1.1). Also, the approach to the boundary of z in conditions (i), (ii) and (1.1) are inside of the unit disc. Instead of these conditions, the behavior of f at the boundary will be considered. In other words, the approach of z from ∂D to points γ will be considered as a condition. Such results were first announced in [2].

Let 𝕌(z, r) be an open disc with centre z and radius r.

We propose the following assertion for the proofs of our theorems:

(A) Let u = u(z) be a positive harmonic function on the open disk 𝕌(z, r_{0}), r_{0} > 0. Suppose that for θ_{0} ∈ [0, 2π), lim_{r→r_{0}} u(re^{iθ_{0}} ) = 0 is satisfied. Then

lim inf_{r→r_{0}} \frac{u(re^{iθ_{0}} )}{r_{0} − r} > 0.

This assertion follows from the Harnack inequality. For more general results and related estimates, see [8, Theorem 1.1], [9], [10].

In addition, after having submitted the present paper, [11] was posed on ResearchGate in which further results are discussed.

2. Main Results

Let d(z, G) be a distance of the point z from the set G .

Theorem 2.1. Let μ ∈ 𝔑, μ_{0} > 3, f be a holomorphic function in the unit disc that is continuous on \overline{D} ∩ 𝕌 (1, δ_{0}) for some δ_{0} > 0 and |f(z) − 1| < 1 for |z| < 1 such that

f(z) = 1 + z + O(μ(|z − 1|)), z ∈ ∂D, z → 1.

Then f(z) = 1 + z.

Proof. Consider the function

φ(z) = f(z) − 1 = 1 + z + O(μ(|z − 1|)) − 1 = z + O(μ(|z − 1|)).

There exists a number c_{1} > 0 such that

|φ(z) − z| ⩽ c_{1}μ(|z − 1|), ∀z ∈ ∂D ∩ 𝕌(1, δ_{0}).

Let us denote k and c_{2} as follows

k = sup_{\begin{matrix}|z−1|=δ_{0} \\ z∈D \end{matrix}} |φ(z) − z|, c_{2} = max\{ \frac{k}{μ(δ_{0})}, c_{1}\}.

It can be easily seen that for all boundary points of the set D ∩ 𝕌(1, δ_{0}), the inequality |φ(z) − z| ⩽ c_{2}μ(|z − 1|) is satisfied. Applying Theorem 3 in [12] (see also [3], [1]) to the set D ∩ 𝕌(1, δ_{0}) and the function φ(z) − z, one receives

(2.1)  |φ(z) − z| ⩽ c_{2}μ(|z − 1|), ∀z ∈ D ∩ 𝕌(1, δ_{0}).

From μ_{0} > 3, we take that there exist some positive constants ε > 0 and σ < min(δ_{0}, 1) such that

\frac{logμ(x)}{logx} ⩾ 3 + ε, ∀x ∈ (0, σ).

Otherwise

(2.2)  μ(x) ⩽ x^{3+ε} , ∀x ∈ (0, σ).

By (2.1) and (2.2), we obtain

(2.3)  |φ(z) − z| ⩽ c_{2}|z − 1|^{3+ε} ∀z ∈ D ∩ 𝕌(1, σ).

Consider the harmonic function h defined as

h(z) = Re( \frac{1 + φ(z)}{1 − φ(z)} ) − Re( \frac{1 + z}{1 − z} ) .

The function

\frac{1 + φ(z)}{1 − φ(z)}

maps the disc D to the right half plane and hence, the first term of h(z) is nonnegative, the second term is zero on ∂D ∖ {1}. Therefore,

(2.4)  liminf_{z→ς,z∈D} h(z) ⩾ 0, ∀ς ∈ ∂D ∖ {1}.

With the simple calculations, we take

h(z) = Re( \frac{2(φ(z) − z)}{(1 − φ(z))(1 − z)} ) .

From (2.3),

lim_{z→1,z∈D} \frac{1 − φ(z)}{1 − z} = 1

and, so there exists δ_{1} ∈ (0, σ) such that

|1 − φ(z)| ⩾ \frac{1}{2}|z − 1| ∀z ∈ D ∩ 𝕌(1, δ_{1}).

Therefore,

|(1 − φ(z))(1 − z)| ⩾ \frac{1}{2}|z − 1|^{2} ∀z ∈ D ∩ 𝕌(1, δ_{1})

and

(2.5) | \frac{2(φ(z) − z)}{(1 − φ(z))(1 − z)} | ⩽ \frac{c_{2}|z − 1|^{3+ε}}{\frac{1}{2}|z − 1|^{2}} = 2c_{2}|z − 1|^{3+ε} ∀z ∈ D ∩ 𝕌(1, δ_{1}).

Consequently,

(2.6)  lim_{z→1,z∈D} h(z) = 0.

Applying the maximum principle [7, p. 48] to the harmonic function h(z), from (2.4) and (2.6), we conclude either h(z) > 0, ∀z ∈ D or h ≡ 0. If h is not a constant, taking z = r in (2.5) gives us

(2.7)  lim_{r→1} \frac{h(r)}{1 − r} = 0.

(2.6) and (2.7) contradict with assertion (A) statement. Consequently, h ≡ 0. This implies that φ(z) = z and f(z) = (1 + z). □

Theorem 2.2. Let ϕ : D → D be a finite Blaschke product which equals τ ∈ ∂D on a finite set A_{f} ⊂ ∂D. Let f be a holomorphic function in the unit disc that is continuous on \overline{D}∩\{z : d(z, A_{f} ) < δ_{0}\} for some δ_{0} > 0 and |f(z)−1| < 1 for |z| < 1. Assume that μ^{1}, μ^{2} ∈ 𝔑, μ_{0}^{1} > 3, μ_{0}^{2} > 2. Suppose that the following conditions are satisfied (i) for a given γ_{0} ∈ A_{f}

(2.8)  f(z) = 1 + ϕ(z) + O(μ^{1}(|z − γ_{0}|)), z ∈ ∂D, z → γ_{0},

(ii) for all γ ∈ A_{f} − {γ_{0}}

(2.9)  f(z) = 1 + ϕ(z) + O(μ^{2}(|z − γ|)), z ∈ ∂D, z → γ.

Then f(z) = 1 + ϕ(z).

Proof. Let Φ(z) = f(z) − 1. By using (2.8) and (2.9), we obtain for a given γ_{0} ∈ A_{f}

(2.10)  Φ(z) = ϕ(z) + O(μ^{1}(|z − γ_{0}|)), z ∈ ∂D, z → γ_{0}

and for all ∀γ ∈ A_{f} − {γ_{0}}

Φ(z) = ϕ(z) + O(μ^{2}(|z − γ|)), z ∈ ∂D, z → γ.

Without loss of generality, we may assume that τ = 1 and γ_{0} = 1. Due to (2.10), there exist numbers c_{3} > 0, δ_{0} ∈ (0, 1) such that

|Φ(z) − ϕ(z)| ⩽ c_{3}μ^{1}(|ς − 1|), ∀ς ∈ ∂D, |ς − 1| < δ_{0}.

Let us denote p and c_{4} as follows;

p = sup_{|z−1|=δ_{0}_{z∈D}} |Φ(z) − ϕ(z)|, c_{4} = max\{ \frac{p}{μ^{1}(δ_{0})}, c_{3}\} .

It can be easily seen that for all boundary points of the set D ∩ 𝕌(1, δ_{0}), the inequality

limsup_{z→ς,z∈D}|Φ(z) − ϕ(z)| ⩽ c_{4}μ^{1}(|ς − 1|)

is satisfied. Applying Theorem 3 of [2] to the set D ∩ 𝕌(1, δ_{0}) and the function Φ(z) − ϕ(z), one yields

(2.11)  |Φ(z) − ϕ(z)| ⩽ c_{4}μ^{1}(|z − 1|), ∀z ∈ D ∩ 𝕌(1, δ_{0}).

From μ_{0} > 3, there are some positive constants ε and σ < min(δ_{0}, 1) such that inequality (2.2) is satisfied. Combining (2.2) and (2.11), we obtain

(2.12)  |Φ(z) − ϕ(z)| ⩽ c_{4}(|z − 1|)^{3+ε} , ∀z ∈ D ∩ 𝕌(1, σ).

Analogously, for any point γ ∈ A_{f} − {1}, from conditions μ_{0}^{2} > 2 and (2.9), we have

(2.13)  |Φ(z) − ϕ(z)| ⩽ c_{5}(|z − γ|)^{2+ε} , ∀z ∈ D ∩ 𝕌(γ, σ_{1})

with some positive constants c_{5} and σ_{1} We introduce the harmonic function

ψ(z) = Re( \frac{1 + Φ(z)}{1 − Φ(z)} ) − Re( \frac{1 + ϕ(z)}{1 − ϕ(z)} ) .

Since a finite Blaschke product ϕ is a holomorphic function throughout \overline{D} and |ϕ| = 1 on ∂D, we have that the second term of ψ is zero on ∂D − A_{f} . The first term of ψ is nonnegative. Consequently,

(2.14)  liminf_{z→ς,z∈D} ψ(z) ⩾ 0, ∀ς ∈ ∂D − A_{f}

Now, let us examine the behavior of the function ψ at points of set A_{f} . After simple calculations, we obtain

ψ(z) = Re( \frac{2(Φ(z) − ϕ(z))}{(1 − Φ(z))(1 − ϕ(z))} ) .

Now, let’s take any point γ ∈ A_{f} − {1}. For the finite Blaschke product ϕ, obviously |ϕ′(z)| > 0 for any z, |z| = 1. If |ϕ′(γ)| = c_{γ}, then there exists a constant σ_{γ} ∈ (0, σ_{1}) such that

(2.15)  |1 − ϕ(z)| ⩾ \frac{c_{γ}}{2} |γ − z| ∀z ∈ D ∩ 𝕌(γ, σ_{γ}).

From (2.13), we get lim_{z→γ} \frac{1−Φ(z)}{1−z} = c_{γ} and there exists σ′_{γ} ∈ (0, σ_{γ}) such that

(2.16)  |1 − Φ(z)| ⩾ \frac{c_{γ}}{2} |γ − z| ∀z ∈ D ∩ 𝕌(γ, σ′_{γ}).

Then, from (2.13), (2.15) and (2.16),

| \frac{2(Φ(z) − ϕ(z))}{(1 − Φ(z))(1 − ϕ(z))} | ⩽ \frac{8c_{5}}{c_{γ}^{2}} ∀z ∈ D ∩ 𝕌(γ, σ′_{γ}).

Thus, the function ψ(z) is bounded below (even bounded) at the some neighborhood of every point γ ∈ A_{f} − {1}.

Using (2.12) (instead of (2.13), analogously we may conclude

(2.17)  |ψ(z)| ⩽ c_{6}|z − 1|^{1+ε} ∀z ∈ D ∩ 𝕌(γ, σ′_{γ})

for some positive constants c_{6} and σ′, from which in particular

(2.18)  lim_{z→1,z∈D} ψ(z) = 0.

So, the harmonic function ψ(z) satisfied condition (2.14) and is bounded below at the some neighborhood of every point of finite set A_{f} . Then, from the Phragmen– Lindelöf principle, [7] we obtain either ψ(z) > 0, ∀z ∈ D or ψ ≡ 0. Taking z = r in (2.17) give us

(2.19)  lim_{z→1} \frac{ψ(r)}{1 − r} = 0.

If ψ is not constant, (2.18) and (2.19) contradict with assertion (A) statement. Consequently, ψ ≡ 0. This implies that Φ(z) = ϕ(z) and f(z) = (1 + ϕ(z)). □

Acknowledgement. We thank the referee for his/her constructive comments and suggestions on the earlier version of this paper.

References

1. T. G. Aliyev, Irregular boundary zeros of analytic functions in contour-solid theorems, Bull. Soc. Sci. Letters Lodz 49 Ser. Rech. Deform. 28 (1999), 45–53.

2. T. Aliyev and B. N. Örnek, A generalized Schwarz lemma on the boundary, Ukrainian Mathematics Congress-2001, Natsional Akad. Nauk Ukraini, Inst. Mat. Kiev, (2003), 5-9.

3. T. G. Aliyev, P. M. Tamrazov, A contour-solid problem for meromorphic functions, taking into account nonunivalence, Ukr. Math. J. 39 (1987), 683–690.

4. D. M. Burns, S. G. Krantz, Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary, J. Am. Math. Soc. 7 (1994), 661–676.

5. D. Chelts, A generalized Schwarz lemma at the boundary, Proc. Am. Math. Soc. 129(11) (2001), 3275–3278.

6. G. M. Golusin, Geometric Theory of Functions of Complex Variable, 2nd ed., Moscow, 1966, (in Russian).

7. W. K. Hayman, P. B. Kennedy, Subharmonic Function, Academic Press, London, 1976.

8. M. Mateljević, The lower bound for the modulus of the derivatives and Jacobian of harmonic injective mappings, Filomat 29(2) (2015), 221–244.

9. , Distortion of harmonic functions and harmonic quasiconformal quasi-isometry, Revue Roum. Math. Pures Appl. 51(56) (2006), 711–722.

10. , Ahlfors–Schwarz lemma and curvature, Kragujevac J. Math. 25 (2003), 155–164.

11. , Note on Rigidity of Holomorphic Mappings & Schwarz and Jack Lemma, (in preparation), ResearchGate.

12. P. M. Tamrazov, Holomorphic functions and mappings in the contour-solid problem, Dokl. Akad. Nauk SSSR 279(1) (1984), 38–43.

Department of Computer Engineering (Received 30 06 2015) Amasya University (Revised 30 08 2015) Merkez-Amasya 05100 Turkey

 nafi.ornek@amasya.edu.tr

Department of Mathematics Gebze Technical University Gebze-Kocaeli Turkey

 tugbaakyel03@gmail.com