Abstract
We obtain an new boundary Schwarz inequality, for analytic functions mapping the unit disk to itself. The result contains and improves a number of known estimates.
1 Introduction
Denote by ∆ ⊂ ℂ the open unit disk, and let f: ∆ → ∆ be analytic. We assume that there is x > ∂∆ and β > ℝ such that
By pre-composing with a rotation we may suppose that x = 1, and by post-composing with a rotation we may suppose that f(1) = 1. Then Julia’s Lemma (e.g. [1, 2]) gives
This inequality has an appealing geometric interpretation, which we do not use here. But two immediate consequences which we do use, are that β > 0 and that the radial derivative of f exists at 1 ∈ > ∂∆:
(There are many other consequences of Julia’s Lemma, the most important being contained in the Julia-Carathéodory Theorems.)
Assuming the normalization f(0) = 0, we evidently have β ≥ 1. But even better, Osserman [3] showed that in this case
(A proof of (3) can also be found in [4], which is motivated by the influential paper [5].) Now Osserman’s inequality was in fact anticipated by Ünkelbach [6], who had already obtained the better estimate
However, [3] also contains a non-normalized version, which reduces to (3) if f(0) = 0, viz.
Since the appearance of Osserman’s paper, a good number of authors have refined and generalized these estimates – as discussed in the next section. The aim here is to provide a different and very elementary approach, which contains and improves many of these modification. But first we recall some results which are of use in the sequel.
The well-known Schwarz’s Lemma, which is a consequence of the Maximum Principle, says that if f: ∆ → ∆ is analytic with f(0) = 0, then
To remove the normalization f(0) = 0, one applies Schwarz’s Lemma to ϕf(a) ∘ f ∘ ϕa where ϕa is the automorphism of ∆ which interchanges a and 0:
This gives the Schwarz-Pick Lemma which says that for f: ∆ → ∆ analytic,
Consequently, the hyperbolic derivative satisfies
It is the Schwarz-Pick Lemma that does most of the work in proving Julia’s Lemma. But another consequence of the Schwarz-Pick Lemma is the following (e.g. [7–9]), which we shall also rely upon.
Lemma 1.1
(Dieudonné’s Lemma). Let f: ∆ → ∆ be analytic, with f(z) = w and f(z1) = w1. Then
where
2 Main result
We remove the dependence on f(0), while improving many estimates which do contain f(0). We shall rely on Dieudonné’s Lemma, the Schwarz-Pick Lemma, and Julia’s Lemma.
Theorem 2.1
Let f: ∆ → ∆ be analytic with f(z) = w and f(1) = 1 as in (1). Then
Proof
Using the easily verified identity
we get, in Dieudonné’s Lemma,
and
then having z1 → 1 along a sequence for which β in (1) is attained, we get
That is,
Now, upon squaring both sides of this inequality, there is some cancellation:
That is,
By the Schwarz-Pick Lemma each side of this last inequality is nonpositive, so isolating β we get (6).
Remark 2.2
Having z → 1 radially in line (8), and using (2), we obtain
From this, and using
Remark 2.3
In Lemma 6.1 of [8] is the estimate
which contains (5), but is quite mild if |z| or |f(z)| is near 1. Anyway,
Remark 2.4
Now take z = 0, so that (6) reads
This may be regarded as an non-normalized version of (4). Indeed, taking also f(0) = 0 recovers (4). This is the same estimate which results from having z = 0 in Theorem 5 of [10]. However, that result (which is arrived at by very nonelementary means) contains f(0) even for z ≠ 0, a deficiency from which Theorem 2.1 does not suffer.
Remark 2.5
Using again
3 Consequences
Cases for which z = w = 0 (i.e. f(0) = 0) are obviously contained in the remarks above, but when this holds we can do a little better, as follows.
Corollary 3.1
Let f: ∆ → ∆ be analytic with f(0) = 0 and f(1) = 1 as in (1). Then
Proof
We introduce f″(0), in standard fashion: Set
Then h is analytic on ∆ with h(0) = 0, and by Schwarz’s Lemma h: ∆ → ∆. Here we have
A calculation using the identity (7) and the assumption (1) gives
Then in (6), i.e. (4), replacing f with h and β with
Inserting (13) and a little tidying yields (12), as desired.
Remark 3.2
Corollary 3.1 improves
which was obtained by Dubinin [13] using a proof which relies directly on (3). (Incidentally, Schwarz’s Lemma applied to h gives |f″(0)|/2 ≤ 1 − |f′(0)|2, from which it is readily seen that (15) improves (3).)
Remark 3.3
We add finally using that (4) in the form
then replacing f with h and β with
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