A SHARP SCHWARZ INEQUALITY ON THE BOUNDARY

ROBERT OSSERMAN

Abstract. A number of classical results reflect the fact that if a holomorphic function maps the unit disk into itself taking the origin into the origin, and if some boundary point b maps to the boundary, then the map is a magnification at b. We prove a sharp quantitative version of this result which also sharpens a classical result of Loewner, and which implies that the map is a strict magnification at b unless it is a rotation.

The standard Schwarz Lemma states that an analytic function f(z) mapping the unit disk into itself, with f(0) = 0, must map each smaller disk |z| < r < 1 into itself and (as a result) satisfy |f′(0)| ≤ 1. Furthermore, unless f is a rotation, one has strict inequality |f′(0)| < 1 and f maps each disk |z| ≤ r < 1 into a strictly smaller one.

It is an elementary consequence of Schwarz’ Lemma that if f extends continuously to some boundary point b with |b| = 1, and if |f(b)| = 1 and f′(b) exists, then |f′(b)| ≥ 1. David Gilbarg raised the question if one again has strict inequality unless f is a rotation. The answer is “yes” (Lemma 1 below), but it does not follow from the standard Schwarz inequality. One needs a stronger form (Lemma 2 below) where one has a quantitative bound on how much each disk |z| ≤ r < 1 is shrunk if f is not a rotation.

Lemma 1 (The boundary Schwarz Lemma). Let f(z) satisfy

(a) f(z) is analytic for |z| < 1,

(b) |f(z)| < 1 for |z| < 1,

(c) f(0) = 0,

(d) for some b with |b| = 1, f(z) extends continuously to b, |f(b)| = 1, and f′(b) exists.

Then

|f′(b)| ≥ \frac{2}{1 + |f′(0)|}. (1)

Corollary 1. Under hypotheses (a) - (d),

|f′(b)| ≥ 1 (2) and

Research at MSRI is supported in part by NSF grant DMS-9701755 .

(3) |f′(b)| > 1 unless f(z) = e^{iα}z, α real.

Proof. Inequalities (2) and (3) follow immediately from (1) together with the standard Schwarz Lemma.

Corollary 2. Let f satisfy conditions (a), (b), (c) of the lemma, and suppose that f extends continuously to an arc C on |z| = 1, with |f(z)| = 1 on C. Then the length s of C and the length σ of f(C) satisfy

σ ≥ \frac{2}{1 + |f′(0)|} s. (4)

Proof. By the reflection principle, f extends to be analytic on the interior of C and therefore satisfies condition (d) of Lemma 1. Hence (4) follows from (1).

Remarks 1. 1. Again by the standard Schwarz Lemma, (4) implies that σ ≥ s, and σ > s unless f is a rotation. That is the content of a classical theorem of Loewner [L]. (See also Velling [V].)

2. The length σ of f(C) is to be taken with multiplicity, if f(C) is a multiple covering of the image.

3. Inequality (1) is sharp, with equality possible for each value of |f′(0)|.

4. One can drop the condition (c) that f(0) = 0. Analogous results hold for any value of f(0). See Lemma 3, the General Boundary Lemma, below.

5. One does not need to assume that f extends continuously to b. For example, if f has a radial limit c at b, with |c| = 1, and if f has a radial derivative at b, then that derivative also satisfies the inequality (1). More generally, if for some b with |b| = 1, there exists a sequence z_{n} such that z_{n} → b and f(z_{n}) → c, with |c| = 1, then

\underline{lim}_{z_{n}→b} |\frac{f(z_{n}) − c}{|z_{n}| − |b|}| ≥ \underline{lim}_{z_{n}→b} \frac{1 − |f(z_{n})|}{1 − |z_{n}|} ≥ \frac{2}{1 + |f′(0)|}. (5)

Both Lemma 1 and the statement about radial limits are immediate consequences, since in either case we may choose z_{n} = t_{n}b for t_{n} real, t_{n} → 1, and the left-hand side of (5) becomes |f′(b)|.

Lemma 2 (Interior Schwarz Lemma). Let f(z) satisfy conditions (a), (b), (c) of Lemma 1. Then

|f(z)| ≤ |z|\frac{|z| + |f′(0)|}{1 + |f′(0)||z|} for |z| < 1. (6)

Proof. Let g(z) = \frac{f(z)}{z}. Then by the standard Schwarz Lemma, either f is a rotation, or else |g(z)| < 1 for |z| < 1. In the former case, |f′(0)| = 1 and (6) holds trivially. So we need only consider the second case, where |g(z)| < 1. Furthermore, since inequality (6) is unaffected by rotations, we may assume that g(0) = f′(0) = a, where 0 ≤ a < 1. Then (6) is equivalent to

|g(z)| ≤ \frac{|z| + a}{1 + a|z|} for |z| < 1, with a = g(0). (7)

But that is an immediate consequence of the standard Schwarz-Pick version of the Schwarz Lemma, which says that g must map each disk |z| < r into the image of that disk under the linear fractional map

G(z) = \frac{z + a}{1 + az}

which is a circular disk whose diameter is the interval

[\frac{a − r}{1 − ar}, \frac{a + r}{1 + ar}]

of the real axis.

Hence,

|z| = r ⇒ |g(z)| ≤ \frac{a + r}{1 + ar} = \frac{|z| + a}{1 + a|z|},

which proves (7), and hence (6).

Remarks 2. 1. For related sharpened forms of the interior Schwarz Lemma, see

Mercer [M].

2. Inequality (7) is sharp, with equality for g(z) = G(z), z = r. Hence, inequality (6) is sharp, with equality for the function

f(z) = z \frac{z + a}{1 + az} , 0 ≤ a < 1,

when z is on the positive real axis. The same function gives equality in (1) when b = 1.

3. When f is not a rotation, (6) is a strict improvement on the standard Schwarz Lemma, since the second factor on the right is strictly less than 1 when |f′(0)| < 1.

Proof of Lemma 1. Let f satisfy conditions (a),(b),(c) of Lemma 1. Then, using the upper bound (6) for |f(z)|, we have for any b and c with |b| = 1, |c| = 1,

|\frac{f(z) − c}{|z| − |b|}| ≥ \frac{1 − |f(z)|}{1 − |z|} ≥ \frac{1 + |z|}{1 + |f′(0)||z|}.

As |z| → 1, the right-hand side tends to \frac{2}{1+|f′(0)|}.

This proves (5), and as noted in Remark 5 above, Lemma 1 follows.

Lemma 3 (The General Boundary Lemma). Under hypotheses (a),(b), and (d) of Lemma 1, one has

|f′(b)| ≥ \frac{2}{1 + |F′(0)|} \frac{1 − |f(0)|}{1 + |f(0)|}, (8)

where F is defined in (9) below and satisfies |F′(0)| ≤ 1, with strict inequality unless F is a rotation and f is an automorphism of the unit disk.

Proof. Let

F (z) = \frac{f(z) − f(0)}{1 − \overline{f(0)}f(z)}. (9)

Then F satisfies the hypotheses of Lemma 1, and therefore

|F′(b)| ≥ \frac{2}{1 + |F′(0)|}, (10)

with equality if and only F is a rotation. But a calculation gives

F′(z) = f′(z)\frac{1 − |f(0)|^{2}}{[1 − \overline{f(0)}f(z)]^{2}} .

Since |f(b)| = 1 implies

|1 − \overline{f(0)}f(b)| ≥ 1 − |\overline{f(0)}f(b)| = 1 − |f(0)|,

we have

|F′(b)| = |f′(b)|\frac{1 − |f(0)|^{2}}{|1 − \overline{f(0)}f(b)|^{2}} ≤ |f′(b)|\frac{1 + |f(0)|}{1 − |f(0)|}. (11)

Combining (10) and (11) yields (8).

Remarks 3 (Concluding Remarks). 1. An interesting special case of Lemma 1 is when f′(0) = 0, in which case inequality (1) implies |f′(b)| ≥ 2. Clearly equality holds for

f(z) = e^{iα}z^{2}, α real. (12)

Furthermore, that is the only case of equality; the same type of argument used to prove Lemmas 1 and 2 yields a stronger inequality that implies |f′(b)| > 2 unless f is of the form (12). More generally, the argument of the standard Schwarz lemma shows that if f(z) = ∑_{n=0}^{∞} a_{n}z^{n} satisfies (a), (b) of Lemma 1 and if

a_{0} = a_{1} = ⋯ a_{k−1} = 0, (13)

then |a_{k}| ≤ 1, and |a_{k}| = 1 if and only if

f(z) = e^{iα}z^{k}, α real. (14)

Furthermore, either (14) holds, or else |f(z)| < |z|^{k} for |z| < 1. The argument of Lemma 2 yields the stronger result that

|f(z)| ≤ |z|^{k} \frac{|z| + |a_{k}|}{1 + |a_{k}||z|}. (15)

Using (15) in the proof of Lemma 1 then shows that if also condition (d) of Lemma 1 holds, then

|f′(b)| ≥ k + \frac{1 − |a_{k}|}{1 + |a_{k}|}. (16)

It follows that |f′(b)| ≥ k, with equality only if f is of the form (14).

2. A corollary of Lemma 3 is that under the same hypotheses, one has

|f′(b)| ≥ \frac{1 − |f(0)|}{1 + |f(0)|} (17)

and the inequality is strict unless f is an automorphism of the unit disk. In this context, see Carathéodory [C1], pp.54-55, on Julia’s Theorem.

3. For related results, and other types of boundary Schwarz Lemmas, see Carathéodory [C2], pp.54-55, on the converse of Julia’s Theorem, Pommerenke [P], p. 71, on the Julia-Wolff Lemma, and the paper of Burns and Krantz [BK].

References

[BK] Burns, D.M. and Krantz, S.G. “Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary.” J. Amer. Math. Soc. 7 (1994) 661-676.

[C1] Carathéodory, C. Conformal Representation Cambridge University Press 1952.

[C2] Carathéodory, C. Theory of Functions Vol.2, New York, Chelsea 1954.

[L] Löwner. K. “Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I.” Math. Ann. 89 (1923) 103-121.

[M] Mercer, P.R. “Sharpened versions of the Schwarz Lemma.” J. Math. Anal. Appl. 205 (1997) 508-511.

[P] Pommerenke, Ch. Boundary Behavior of Conformal Maps, New York, Springer-Verlag 1992.

[V] Velling, J.A. “The uniformizations of rectangles, an exercise in Schwarz’s Lemma.” Amer. Math. Monthly 39 (1992) 112-115.

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