1. Introduction

One of the very basic theorems in complex analysis is the following Schwarz lemma [4].

Theorem 1.1 (Schwarz Lemma). Let f be a holomorphic function of the open unit disc U = {z ∈ ℂ : |z| ＜ 1} into U with f(0) = 0. Then |f′(0)| ≤ 1 and |f(z)| ≤ |z| for all z ∈ U with the equality only for f(z) = eiθz with real θ.

If, in addition, f has multiple zeroes at z = 0, then the Schwarz lemma results in the following (see [7]).

Corollary 1.2.  Let f be a holomorphic function of U into U with f(0) = f′(0) = … = f(n−1)(0) = 0. Then |f(z)| ≤ |z|n for all |z| ＜ 1 with the equality only for f(z) = eiθzn with real θ.

More generally, the Schwarz lemma can be applied to a function with the information f(α) = β for some α, β with |α| ＜ 1, |β| ＜ 1 instead of f(0) = 0 and it is called the Schwarz-Pick lemma [2].

Corollary 1.3 (Schwarz-Pick Lemma). Let f be a holomorphic function of U into U with f(α) = β for some α, β with |α| ＜ 1, |β| ＜ 1. Then

for |α| ＜ 1.

If f in Corollary 1.3 fixes α, then |f′(α)| ≤ 1. Note that the equality in the above corollary holds only for Möbius transformation mapping the open unit disc into itself.

The Schwarz lemma looks a simple result, but it is highly influential in the function theory of the complex analysis. It is used to get properties of holomorphic functions of the unit disc into itself at a boundary point of the unit disc. For historical background about the Schwarz lemma and its applications on the boundary of the unit disc, we refer to Boas [1]. In [3], we find a holomorphic self map defined on the closed unit disc with fixed points only on the boundary of the unit disc.

Now, our concern is for holomorphic functions mapping the unit disc into itself at a boundary point of the unit disc. From the Schwarz lemma, it is known that if a holomorphic function f of 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. The following theorem called the boundary Schwarz lemma can be found in [6].

Theorem 1.4 (The boundary Schwarz Lemma). Let f be a holomorphic function of U into U with f(0) = 0. Assume that for some point z0 with |z0| = 1, f extends continuously to z0, |f(z0)| = 1, and f′(z0) exists. Then

and hence

The equality in (1.2) holds if and only if f(z) = zeiθ for some real θ.

If, in addition, the function f has the property f(0) = f′(0) = … = f(n−1)(0) = 0, n ∈ ℕ, then

The equality in (1.3) holds if and only if f(z) = zneiθ for some real θ.

Remark 1.5. Under the same hypothesis as in Theorem 1.4 except f(0) = 0, Osserman [6] showed that the following inequality

holds where F(z) = instead of the inequality (1.1).

The assumption in Theorem 1.4 that f extends continuously to z0 with |z0| = 1, |f(z0)| = 1, and f′(z0) exists can be changed to the assumption that f has a radial limit w0 at z0 with |z0| = 1, |w0| = 1, f has a radial derivative at z0. (see [6]).

Recently, Örnek [5] proved the following inequality at a boundary point of the unit disc.

Theorem 1.6.  Let f be a holomorphic function in U with f(0) = 1 and |f(z)−∊| ＜∊ for |z| ＜ 1 and 1/2 ＜∊ ≤ 1. If for some z0 with |z0| = 1, f has an angular limit f(z0) at z0, f(z0) = 2∊, then

Moreover, the equality in (1.5) holds if and only if

for a real θ.

In this paper, we show some inequalities at a boundary point for different form of holomorphic functions and find the condition for equality.

 

2. The Schwarz Lemma and its Application at a Boundary Point

The Schwarz lemma means that any holomorphic function of the unit disc into itself with zero fixed maps each disc centered at zero into a smaller one. Moreover it maps each disc centered at zero into a strictly smaller one if it is not a rotation. From now on, more generally we consider a holomorphic function with zero not fixed. Örnek [5] considered a holomorphic function f on U with f(0) = 1, |f(z) − 𝜖| ＜ 𝜖 where 𝜖 ＞ 1/2. We consider a different form of holomorphic functions and get the following proposition by the similar method.

Proposition 2.1. Let f be a holomorphic function on U satisfying |f(z) − 1| ＜ 1 with f(0) = a where 0 ＜ a ＜ 2. Then, f satisfies the inequality

for |z| ＜ 1. Moreover,

The equality in (2.1) for some nonzero z ∈ U or in (2.2) holds if and only if

for some real θ.

Proof. Let g(z) = f(z) − 1 and let

for z ∈ U.

Then g and w are holomorphic functions on U with |g(z)| ＜ 1 and |w(z)| ＜ 1 for |z| ＜ 1 and w(0) = 0. Hence w satisfies the condition for the Schwarz lemma.

By the Schwarz lemma, |w(z)| ≤ |z| for |z| ＜ 1. Hence,

It implies that

Therefore, we have the inequality (2.1).

On the other hand, the facts that

and |w′(0)| ≤ 1 by the Schwarz lemma induce that

Hence, |f′(0)| ≤ 1 − (a − 1)2 = a(2 − a).

The equality in (2.1) for some nonzero z ∈ U or in (2.2) holds if and only if

that is,

for some real θ.                                    ⧠

Remark 2.2. From the formula (2.5),

|f(z) − a| = |f(z)| − a and |(1 − a)f(z) + a| = |(1 − a)f(z)| + a

should be satisfied at some nonzero point z ∈ U that the equality in (2.1) holds, i.e., 0 ＜ a ＜ 1 and f(z) is real with a ＜ f(z) ＜ 2 at the above nonzero point z ∈ U.

Therefore, if f is the function in (2.3) and 0 ＜ a ＜ 1, then at some nonzero point z ∈ U satisfying a ＜ f(z) ＜ 2 the equality in (2.1) holds.

Theorem 2.3.  Let f be a holomorphic function on U satisfying |f(z)−1| ＜ 1 with f(0) = a where 0 ＜ a ＜ 2. Assume that for some z0 with |z0| = 1, f extends continuously to z0, f(z0) = 2, and f′(z0) exists. Then

The equality in (2.6) holds if and only if

where some real θ satisfies that eiθ = 1/z0.

Proof. Let w be the function in (2.4). Then w′(z) satisfies that

The inequality |w′(z0)| ≥ 1 in (1.2) implies that

If |f′(z0)| = , then |w′(z0)| = 1 and so by Theorem 1.4, w(z) = zeiθ for some real θ. It means that

for some real θ. By the condition f(z0) = 2,

a(1 + z0eiθ) = 2{1 − (1 − a)z0eiθ}.

Hence, z0eiθ = 1 and so θ satisfies that eiθ = 1/z0. Conversely, for the given function

where θ satisfies that eiθ = 1/z0, the equality in (2.6) holds.                                ⧠

Remark 2.4. The inequalities in Proposition 2.1 and Theorem 2.3 with a = 1 coincide with the inequalities in Theorem 1.6 and a theorem in Örnek [5] with 𝜖 = 1.

Now, we consider a holomorphic function f with f(0) − a = f′(0) = … = f(n−1)(0) = 0. A function given by

f(z) = a + cnzn + cn+1zn+1 + … , n ∈ ℕ

with cn ≠ 0, is such a holomorphic function. If we change the role |w′(z0)| ≥ 1 by |w′(z0)| ≥ n in the proof of Theorem 2.3, then the following corollary holds.

Corollary 2.5. Let f be a holomorphic function defined on U by

f(z) = a + cnzn + cn+1zn+1 + … , n ≥ 1

satisfying |f(z) − 1| ＜ 1 on U where 0 ＜ a ＜ 2 and cn ≠ 0. Assume that for some z0 with |z0| = 1, f extends continuously to z0, f(z0) = 2, and f′(z0) exists. Then

The equality in (2.8) holds if and only if

where θ satisfies eiθ = 1/.

Proof. By using the formula of w in (2.4),

where bn, bn+1, … ∈ ℂ and bn = cn/{1 − (a − 1)2} ≠ 0.

Hence, w(0) = w′(0) = … = w(n−1)(0) = 0. By Theorem 1.4 and (2.7),

holds and we get (2.8).                                             ⧠

The following theorem provides a refined inequality at a boundary point z0 than Theorem 2.3. If we apply the inequality (2.2), we find that the following inequality (2.10) implies the inequality (2:6).

Theorem 2.6. Let f be a holomorphic function on U satisfying |f(z)−1| ＜ 1 with f(0) = a where 0 ＜ a ＜ 2. Assume that for some z0 with |z0| = 1, f extends continuously to z0, f(z0) = 2, and f′(z0) exists. Then

The equality in (2.10) holds for the function

with z0 = 1 where b = and a = 1, i.e.,

with z0 = 1 where 0 ≤ b = |f′(0)| ≤ 1.

Proof. Let w be the function in (2.4). By applying the inequality (1.1) to w′(z) and the equation (2.7),

Since,

we have

Hence,

For the equality in (2.10), choose arbitrary b satisfying 0 ≤ b ≤ 1 and let

Then,

and it implies that

Therefore the equality in (2.10) holds at z0 = 1 where b = .

On the other hand,

In order to satisfy that f(1) = 2, the condition a = 1 should be satisfied. So,

where b = = |f′(0)| ≤ 1.                                             ⧠

Remark 2.7. The function f in (2.11) can be represented by the following interesting Mclaurin series at zero.

where 0 ≤ b = |f′(0)| = f′(0) ≤ 1 for |z| ＜ 1.

For example, if b = 0, then f(z) = 1 + z2. If b = 1, then f(z) = 1 + z. If b = , then f(z) = = 1 + z + z2 + … .