Schwarz-Pick-type estimates for the hyperbolic derivative
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2006
0 Peter R. Mercer: Department of Mathematics, Buffalo State College , NY 14222, USA E-mail address:
We denote by Δ the open unit disk in C, and for z ∈ Δ, we denote by φz ∈ Aut(Δ) the automorphism which interchanges 0 and z: φz(λ) = (z − λ)/(1 − zλ). We denote by ρ the hyperbolic distance on Δ:
1. Preliminaries
The following is a well-known consequence of the maximum principle.
to obtain the following.
g(0) = 0,
g (0) =
f (z) 1 − |z|2
1 − f (z) 2
Consequently, f ∗(z) := g (0) has | f ∗(z)| ≤ 1, and so ρ( f ∗(z), ·) is defined on Δ, as
long as f is not an automorphism—for in this case, | f ∗| ≡ 1. As such, we are interested
in the following two results.
ρ 0, f ∗(λ) − ρ 0, f ∗(z)
So, for example, if f ∗(λ) and f ∗(z) are on the same side of a ray emanating from the origin,
then ρ( f ∗(λ), f ∗(z)) ≤ 2ρ(λ, z).
Theorem 1.4 (see [1]). Let f : Δ → Δ be analytic, not an automorphism, with f (0) = 0.
Then
In the next section of this paper, we employ a procedure which yields simple proofs of
Theorems 1.3 and 1.4 and extends these results. In particular, Theorem 1.4 is not
applicable if f (0) = 0, as the function exp((λ + 1)/(λ − 1)) shows. Below however, we obtain
a version (Proposition 2.3) which removes the normalization and applies at any pair of
points in Δ, thus furnishing a more complete analog of Schwarz-Pick Lemma 1.2 for f ∗.
In the final section, we obtain some further related results.
We will use the following easily verified facts.
(A) Schwarz-Pick Lemma 1.2 and a little manipulation reveal that f (λ) lies in the
closed disk with center c = f (z)(1 − |φz(λ)|2)/(1 − | f (z)|2|φz(λ)|2) and radius
r = |φz(λ)|(1 − | f (z)|2)/(1 − | f (z)|2|φz(λ)|2). Consequently, |c| − r ≤ | f (λ)| ≤
|c| + r. That is,
f (z) + φz(λ)
≤ f (λ) ≤ 1 + f (z) φz(λ)
(y + x)/(1 + yx) + x
1 + (y + x)/(1 + yx) x
(y + x)/(1 + yx) + x
÷ 1 − 1 + (y + x)/(1 + yx) x
(y − x)/(1 − yx) − x
1 − (y − x)/(1 − yx) x
(y − x)/(1 − yx) − x
÷ 1 − 1 − (y − x)/(1 − yx) x
2. Results
We see below that the following has Theorem 1.3 as a consequence.
f ∗ z1
/ 1 − f ∗ z1
1 −
f ∗ z1
/ 1 − f ∗ z1
≤ f ∗ z2
≤ 1 +
w2 − w1 1 − z2z1 ,
g z1 = z2 − z1 1 − w2w1
w2 − w1 1 − z2z1 ,
h z2 = z2 − z1 1 − w2w1
g z2 = f ∗ z2 ,
h z1 = f ∗ z1 .
The estimates in (A) give
≤ g z2
≤ g z2
Applying estimates (A) to |h(z2)| now (and observing (B)), we obtain the desired result.
Remark 2.2. If f is not an automorphism, then we may apply the increasing function
t → (1/2) log ((1 + t)/(1 − t)) to either side of Proposition 2.1, and we use (C) and (D) to
obtain
σ1 = g z1 = wz22 −− zw11 11−−wz22zw11 ,
1 −
φσ1 (σ2) − φz1 (z2) / 1 − φσ1 (σ2)
φσ1 (σ2) − φz1 (z2) / 1 − φσ1 (σ2)
Again applying the increasing function t → (1/2) log((1 + t)/(1 − t)) when f is not an
automorphism, we obtain the following, which improves Theorem 1.4. (Having z2 = 0
and requiring f (0) = 0 yield σ1 = σ2.)
≤ p z2
≤ 1 +
= 1 +
f ∗(z)
1 − f ∗(z) 2
Below we refine this result using the same sort of procedure as above. (Then, in
principle, a sharpening of Theorem 1.3 could be obtained via integration.)
Remark 2.4. We cite [3], which contains various other generalizations of Theorem 1.4,
one of which (Corollary 4.4) has conclusion
ρ 1 − z1z2 f ∗ z1 , 1 − w1w2 f ∗ z2
z1z2 − 1 w1w2 − 1
([3] also contains some Euclidean versions, as does [5].)
3. Other results
Theorem 1.3 is obtained in [6] by integrating the following theorem.
f ∗(z)
z 1 + φ f ∗(z) φ f (z) f (0) /z
1 − f ∗(z) 2
Proof. With f as given, set
g(λ) = φ f (z) ◦ f ◦ φz(λ) ,
|h(z)| + |z|2 .
h (0) ≤ |z| 1 + |h(z)|
Now h (0) = g (0)/2(|g (0)|2 − 1), and so
g (0) |h(z)| + |z|2 .
2 1 − g (0) 2 ≤ |z|(1 + |h(z)|)
g (0) = 2 1 − |z|2
f ∗(z) ,
Remarks 3.3. (i) Schwarz’s Lemma 1.1 applied to h gives (|φ f ∗(z)(φ f (z)( f (0))/z)| + |z|2)
/|z|(1 + |φ f ∗(z)(φ f (z)( f (0))/z)|) ≤ 1, so this is indeed a refinement. (ii) The lower estimate
in (A) would similarly yield a lower estimate for |d/dz| f ∗(z)||. We leave the details to
the reader. (iii) In [6], the author compares Theorem 3.1 with Schwarz-Pick Lemma 1.2.
Proposition 3.2 may be similarly compared with Dieudonne´’s lemma (e.g., [2, 4]), which
refines Schwarz-Pick Lemma 1.2. A perfect analog of Dieudonne´’s lemma would read
|d/dz| f ∗(z)|| ≤ ((| f ∗(z)| + |z|2)/|z|(1 + | f ∗(z)|))((1 − | f ∗(z)|2)/(1 − |z|2)) (for
f ∗(0) = 0). However, this is not a refinement: for f (λ) = λ2, we have |d/dz| f ∗(z)|| =
(1 − | f ∗(z)|2)/(1 − |z|2) but (| f ∗(z)| + |z|2)/|z|(1 + | f ∗(z)|) = 2 when z = 0. (At any z
for which f (z) = f (0), we have |h(z)| = | f ∗(z)|, so a perfect analog does occur at such
points.)
Acknowledgments
The author is grateful to George T. Hole, his colleague in the Department of
Philosophy, for bringing [1] to his attention, and to John Pfaltzgraff of The University of North
Carolina at Chapel Hill for bringing [6] to his attention.
[1] A. F. Beardon , The Schwarz-Pick lemma for derivatives , Proceedings (...truncated)