Denote by ∆ ⊂ ℂ the open unit disk, and let f: ∆ → ∆ be analytic. We assume that there is x > ∂∆ and β > ℝ such that
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  showed that in this case
(A proof of (3) can also be found in , which is motivated by the influential paper .) Now Osserman’s inequality was in fact anticipated by Ünkelbach , who had already obtained the better estimate
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.
(Dieudonné’s Lemma). Let f: ∆ → ∆ be analytic, with f(z) = w and f(z1) = w1. Then
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.
Let f: ∆ → ∆ be analytic with f(z) = w and f(1) = 1 as in (1). Then
Using the easily verified identity
we get, in Dieudonné’s Lemma,
then having z1 → 1 along a sequence for which β in (1) is attained, we get
Now, upon squaring both sides of this inequality, there is some cancellation:
By the Schwarz-Pick Lemma each side of this last inequality is nonpositive, so isolating β we get (6).
From this, and using , follows the rather comforting fact that the right-hand side of (6) tends to β as z → 1 radially.
In Lemma 6.1 of  is the estimate
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 . 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.
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.
Let f: ∆ → ∆ be analytic with f(0) = 0 and f(1) = 1 as in (1). Then
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
Corollary 3.1 improves
which was obtained by Dubinin  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).)
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About the article
Published Online: 2018-10-19
Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1140–1144, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0096.
© 2018 Mercer, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0