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Normal families and linear differential equation

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Norbert Steinmetz at TU Dortmund University
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Abstract

It is proved that any family of analytic functions with spherical derivative uniformly bounded away from zero ist normal.
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arXiv:1102.3104v1 [math.CV] 15 Feb 2011
Normal Families
And Linear Differential Equations
by Norbert Steinmetz
Technical University of Dortmund
By Marty’s Criterion (see Ahlfors [1]), normality of any family Fof meromorphic
functions on some domain Dis equivalent to local boundedness of the corresponding
family F#of spherical derivatives
f#=|f′|
1 + |f|2.
Recently, J. Grahl and S. Nevo(1) proved a normality criterion involving the spher-
ical derivative by utilising the so-called Zalcman Lemma (see L. Zalcman [7]). At
first glance it looks very surprising since it is based on a lower bound for the spher-
ical derivative.
Theorem (Grahl & Nevo [3]). Suppose all functions of the family Fsatisfy
f#(z)≥ǫfor some fixed ǫ > 0.Then Fis normal.
The aim of this note is to give a completely different proof, which has the ad-
vantage to yield explicit upper bounds for f#. It is based on a property equivalent
to f#(z)>0, namely local univalence of the function f, which again is equivalent
to the fact that the corresponding Schwarzian derivative
Sf=f′′
f′′
−1
2f′′
f′2
is holomorphic on D.
Theorem. Let fbe meromorphic on the unit disc Dsatisfying f#(z)≥ǫ > 0.
Then fhas the form
(1) f=w1
w2
,
where the functions w1and w2are holomorphic on Dand satisfy
(2) |w1(z)|2+|w2(z)|2≤1
ǫ,
w1w2
w′
1w′
2
= 1,and
w1w2
w′′
1w′′
2
= 0.
Moreover,
(3) f#(z)≤2/ǫ
(1 − |z|)2and |Sf(z)| ≤ 4/ǫ
(1 − |z|)3
hold on D.
Proof. Since f#is non-zero, fis locally univalent and its Schwarzian derivative is
holomorphic on D. It is well known that this implies the representation (1), where
w1and w2form a fundamental set of the linear differential equation
(4) w′′ +1
2Sf(z)w= 0.
Then the third condition in (2) always holds (reflecting the fact that the coefficient
of w′vanishes identically), hence the Wronskian of any two solutions is constant.
1I learned about this in a talk given by J. Grahl at the second Bavarian-Qu´ebec Mathematical
Meeting & Tag der Funktionentheorie, November 22-27, 2010, University of W¨urzburg.
1
2
To make some definite choice we normalise by the second condition in (2), which
makes the pair (w1, w2) unique up to sign and from which
f′=−1
w2
2
and f#=1
|w1|2+|w2|2,
hence the first condition in (2) follows. To prove (3) we just remark that from
w1w2
w′
1w′
2
= 1 and the Cauchy-Schwarz inequality follows
f#=1
|w1|2+|w2|2≤ |w′
1|2+|w′
2|2,
while 1
2Sf=
w′
1w′
2
w′′
1w′′
2
yields
1
2|Sf| ≤ |w′
1||w′′
2|+|w′′
1||w′
2|.
The standard Cauchy estimate
|w| ≤ 1/√ǫ⇒ |w′| ≤ 1/√ǫ
1− |z|and |w′′| ≤ 1/√ǫ
(1 − |z|)2
then gives the estimate in both cases of (3).
Remarks and Questions.
•For ǫ > 0 fixed, the family Fǫof all functions fsatisfying f#≥ǫ, and also the
family SFǫof corresponding Schwarzian derivatives is compact, and
Φǫ(r) = sup{f#(z) : |z| ≤ r, f ∈ Fǫ} ≤ 2ǫ−1(1 −r)−2(0 ≤r < 1)
holds. To obtain a lower bound for Φǫwe consider f(z) = 1 + z
1−ziλ (Hille’s
example [4] showing that Nehari’s univalence criterion [5] is sharp). It has spherical
derivative f#(z) = λ
|1−z2|
2
|f(z)|+|f(z)|−1and Schwarzian derivative Sf(z) =
2(1 + λ2)(1 − |z|2)−2, and satisfies f#(z)> f #(±i) = λ/ cosh π
2λand f#(x) =
λ(1 −x2)−1(−1< x < 1),from which
Φǫ(r)≥log(1/ǫ) + O(log log(1/ǫ))(1 −r)−1(0 < ǫ < ǫ0,0< r < 1)
follows (ǫ0≈0.42 is the maximum of λ/ cosh π
2λin 0 < λ < ∞). The true
value of Φǫ(r) has to remain open; is it C(ǫ)(1 −r)−1? The problem to determine
supFǫ|Sf(z)|also remains open.
•Since |w1|2+|w2|2is subharmonic, the spherical derivative satisfies the mini-
mum principle: If fis meromorphic on some domain D, then f#has no local
minima except at the critical points of f(see also [3], Prop. 4.) Actually, −log f#
is subharmonic off the zeros of f#.
•By Thm. 3 of [3], Fǫ=∅if ǫ > 1/2, while F1/2consists of the rotations of
the Riemann sphere. Using
w1w2
w′
1w′
2
= 1 and 1/f#=|w1|2+|w2|2≥2|w1||w2|,
this may be shown as follows (similar to [3]): We first suppose w1(0) = 0 and set
v(z) = w2(z)w1(z)/z. Then v(0) = w′
1(0)w2(0) = −1, hence max
|z|=r|v(z)| ≥ 1 holds
by the maximum principle, this implying min
|z|=r|z|f#(z)≤1/2 and inf
Df#(z)≤1/2.
3
Also inf
Df#(z) = 1/2 gives |v(z)| ≤ 1 = |v(0)|, thus v(z)≡v(0) = −1. This,
however, is only possible if w1(z) = cz and w2(z) = −1/c, hence f(z) = c2z, and
from inf
Df#(z) = |c|2/(1 + |c|4) then follows |c|= 1. Without the normalisation
f(0) = 0, inf
Df#= 1/2 implies that fis a rigid motion of the sphere.
•The upper bound for f#may be slightly improved. Given z∈Dwe may assume
f(z) = 0 by rotating the Riemann sphere. We then have f#(z) = |w2(z)|−2and
w′
1(z)w2(z) = −1, hence f#(z) = |w′
1(z)|2. By the Schwarz-Pick lemma (thanks to
J. Grahl for the keyword) applied to √ǫw1we thus obtain
f#(z) = |w′
1(z)|2≤1/ǫ
(1 − |z|2)2.
•The representation (1) together with the first condition in (2) implies that f
has bounded Nevanlinna characteristic (fis called of bounded type), so that by the
Ahlfors-Shimizu formula
lim
r→1T(r, f) = 1
πZ1
0
(1 −ρ)Z2π
0
f#(ρeiθ)2dθ dρ
is finite (see Nevanlinna [6]). We note, however, that there are functions of bounded
type having spherical derivative growing arbitrarily fast, see [2]. Thus, although
normal functions [satisfying f#(z) = O((1 − |z|)−1) as |z| → 1] have Nevanlinna
characteristic T(r, f) = O(−log(1 −r)), there are functions of bounded type that
are not normal.
References
[1] L. V. Ahlfors, Complex Analysis, McGraw-Hill 1979.
[2] R. Aulaskari and J. R¨atty¨a, Nevanlinna class contains functions whose sperical derivatives
grow arbitrarily fast, Ann. Acad. Sci. Fenn. Math. 34, 387 - 390 (2009).
[3] J. Grahl and S. Nevo, A note on spherical derivatives and normal families, arXiv: 1010.4654v1
[math.CV], 12 p. (2010).
[4] E. Hille, Remarks on a paper by Zeev Nehari, Bull. Amer. Math. Soc. 55, 552 - 553 (1949).
[5] Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55,
544-551 (1949).
[6] R. Nevanlinna, Eindeutige analytische Funktionen, Springer 1936.
[7] L. Zalcman, A heuristic principle in function theory, Amer. Math. Monthly 82, 813-817
(1975).
Adress: Institut f¨ur Mathematik, D-44221 Dortmund, Vogelpothsweg 87, Germany
E-mail: stein@math.tu-dortmund.de
Homepage: www.mathematik.tu-dortmund.de/steinmetz/
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A note on spherical derivatives and normal families, arXiv: 1010.4654v1 [math
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  • J Grahl
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J. Grahl and S. Nevo, A note on spherical derivatives and normal families, arXiv: 1010.4654v1 [math.CV], 12 p. (2010).