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arXiv:1210.7920v3 [math.DS] 28 Feb 2013
A NOTE ON SCHWARZIAN DERIVATIVES AND NORMAL FAMILIES
DINESH KUMAR AND SANJAY KUMAR
Abstract. We establish a criterion for local boundedness and hence normality of a family
Fof analytic functions on a domain Din the complex plane whose corresponding family of
derivatives is locally bounded. Furthermore we investigate the relation between domains
of normality of a family Fof meromorphic functions and its corresponding Schwarzian
derivative family. We also establish some criterion for the Schwarzian derivative family
of a family Fof analytic functions on a domain Din the complex plane to be a normal
family.
1. Introduction
A family Fof analytic functions on a domain Ω ⊂Cis locally bounded on Ω if it is
uniformly bounded on each compact subset of Ω.Equivalently, Fis locally bounded on Ω
if it is uniformly bounded in a neighborhood of each point of Ω (see [1], [4]). By Marty’s
theorem (see [1]), a family Fof meromorphic functions on a domain Ω ⊂Cis normal if
and only if the family F#={f#:f∈F}of the corresponding spherical derivatives is
locally uniformly bounded in Ω,where f#=|f′|
1+|f|2.
It is well known [4], for a family Fof locally bounded analytic functions on a domain
D, the corresponding family of derivatives F′={f′:f∈F}also forms a locally bounded
family in D. Its converse is not true in full generality, however some partial converse [4]
holds. We also establish a partial converse to this result. To our best knowledge, it hasn’t
been studied so far what can be said about the relation between the domains of normal-
ity of a family Fof meromorphic functions, its conjugate family and the corresponding
Schwarzian derivative (SD for short) family. The aim of the present paper is to tackle
these questions. We also establish a criterion for the SD family of a family Fof analytic
functions on a domain D⊂Cto be a normal family.
For a meromorphic function fon C, its SD is defined by Sf(z) = f′′′(z)
f′(z)−3
2(f′′(z)
f′(z))2,
which has the invariance property Sτ◦f=Sffor every Mobius transformation τ(z) =
az+b
cz+d, ad −bc 6= 0.It is well known [3] Sf= 0 if and only if fis a Mobius transformation.
Also if fis a meromorphic function on Cand f(z)6= 0 for all z∈C,then Sf=S1
f.We show
this fact is true even if the meromorphic function fomits a finite value. For meromorphic
2010 Mathematics Subject Classification. 37F10, 30D05.
Key words and phrases. Schwarzian derivative, normal family, meromorphic function, locally bounded.
The research work of the first author is supported by research fellowship from Council of Scientific and
Industrial Research (CSIR), New Delhi.
1
2 D. KUMAR AND S. KUMAR
functions fand gfor which the composition g◦fis defined, Sg◦f= (Sg◦f)f′2+Sfholds.
As Sτ= 0 for each Mobius transformation τ , Sg◦τ= (Sg◦τ)τ′2.
Recently, Steinmetz [5] gave a completely different proof of a normality criterion involv-
ing spherical derivatives given by Grahl and Nevo [2]. The proof was based on a property
equivalent to f#(z)>0,which is again equivalent to the fact that corresponding SD is
holomorphic on the given domain.
2. Theorems and their proofs
Theorem 2.1. Let Fbe a family of analytic functions on the unit disk Dsuch that f(0) = 0
for all f∈F.Suppose F′is locally bounded on D.Then the family Fis locally bounded on
D.
Proof. As F′is locally bounded, therefore Fis equicontinuous on each compact subset of
D.We now show Fis pointwise bounded on Dwhich will imply Fis locally bounded on
D.Let z∈Dbe arbitrary. For each f∈Fintegrating falong the straight line segment L
joining 0 to zwe have
|f(z)−f(0)|=|Zz
0
f′(z)dz|
≤K|z|
for some K > 0 depending on the line segment L. Now consider any disk about the point
zand integrating along any straight line segment in the disk one gets a similar inequality
and the result follows.
Remark 2.2.The above result is true for any arbitrary domain D⊂Cand for any z0∈D
which is common fixed point of the analytic family F.
We now establish a theorem which gives a criterion for the SD family of a family Fof
analytic functions on a domain D⊂Cto be a normal family.
Theorem 2.3. Let Fbe a family of analytic functions on a domain D⊂Csuch that the
family of its derivatives F′satisfies:
(1) |f′| ≥ ǫfor all f∈Fand for some fixed ǫ > 0
(2) The family F′is locally bounded.
Then the SD family is normal on D.
Proof. |f′|>0 is equivalent to local univalence of the function f. As the derivative of
a locally bounded family of analytic functions is locally bounded (see [1]), therefore the
families F′′ and F′′′ are locally bounded. For f∈F, Sf(z) = f′′′ (z)
f′(z)−3
2(f′′(z)
f′(z))2and using the
local boundedness of the families F′′ and F′′′,one obtains the SD family is locally bounded
on Dand so is normal on Dby Montel’s theorem.
NORMALITY AND SCHWARZIAN DERIVATIVES 3
We now show if a meromorphic function fomits a finite value then Sf=S1
f.
Theorem 2.4. For a meromorphic function fon Cwhich omits a finite value, Sf=S1
f.
Proof. Suppose fomits a value w∈C.Define a new function g(z) = f(z)−w. Then g
omits 0. Also Sg=S1
g, Sg=Sf−w=Sfand S1
g=S1
f−w=S1
f,the result follows.
We wanted to analyse the relation between the domains of normality of a family Fof
meromorphic functions on the complex plane Cand that of its SD family. We observe as
such no correlation between them and tried to substantiate this through multiple examples.
In all the subsequent examples to follow the domain of the family of meromorphic functions
under consideration is the complex plane C.
Example 2.5.fn(z) = enz .Its domain of normality is C\ {z:Re(z) = 0}.The SD family
is Sfn(z) = −n2
2which is normal in C.
Example 2.6.fn(z) = ez
nz+1 .The domain of normality of this family is the complex plane
C.The SD family is Sfn(z) = −1
2(nz+1)4which is normal in the punctured complex plane
C\0.
However there is possibility of similar domains of normality as shown by the following
example.
Example 2.7.fn(z) = ez−n. Domain of normality is C.The SD family is Sfn(z) = −1
2
which is also normal in C.
Under some condition on the family Fof meromorphic functions, we show its domain
of normality is contained in the domain of normality of the corresponding SD family. We
will need the following lemmas .
Lemma 2.8. [3] Consider a family Fof locally injective meromorphic functions on a do-
main D⊂Cwhich converges locally uniformly to a locally injective meromorphic function
fon D, then the corresponding SD family of Fconverges locally uniformly on Dto Sf.
Proof. Using the local injectiveness of both the family Fand the limit function fand local
uniform convergence of Fon Dto f, we get the corresponding SD family of Fconverges
locally uniformly on Dto Sfand the result gets proved.
Lemma 2.9. [4] Let Fbe a normal family of analytic functions on a domain D⊂C.If
for all f∈F,|f(ζ)| ≤ L, for some point ζ∈Dand for some L < ∞,then Fis locally
bounded.
Theorem 2.10. Let Fbe a family of locally injective meromorphic functions on C. Let D
be the domain of normality of F. Then Dis contained in the domain of normality of the
corresponding SD family if the limit function of Fon Dis locally injective.
4 D. KUMAR AND S. KUMAR
Proof. Let fbe the limit function of Fon D. Using Lemma 2.8, the SD family of F
converges locally uniformly on Dto Sf. By definition Dis contained in the domain of
normality of the SD family.
Theorem 2.11. Let Fbe a family of analytic functions on C.Let Dbe the domain of
normality of F.Suppose Fsatisfies following conditions on D:
(1) For some point ζ∈Dand for all f∈F,|f(ζ)| ≤ Lfor some L < ∞,
(2) |f′| ≥ ǫfor all f∈Fand for some fixed ǫ > 0
Then Dis contained in the domain of normality of the corresponding SD family of F.
Proof. Using Lemma 2.9 Fis locally bounded and from Theorem 2.3 the SD family is
locally bounded on Dand so is normal on Dby Montel’s theorem. The result then
follows.
We will require the following definition.
Definition 2.12.Let fand gbe two meromorphic functions on the complex plane C.Then
fis conjugate to gdenoted by f∼g, if there exists a Mobius transformation φsatisfying
φ◦f=g◦φ.
Analogously one defines two families Fand Gof meromorphic functions to be conjugate if
there exist a family Aof Mobius transformations satisfying φ◦f=g◦φfor all f∈F, g ∈G
and φ∈A.
We further analyse the relationship between the domain of normality of a family of
meromorphic functions and that of its conjugate family. The resultant is no correlation
justified with examples.
Example 2.13.Consider fn(z) = enz and gn(z) = nezfor all n∈N.Then fn∼gnunder
φn(z) = nz, n ∈N.The domain of normality of the former is C\ {z:Re(z) = 0}while
that of the latter is C.
There is possibility of similar domains of normality as exhibited by the following example.
Example 2.14.Consider fn(z) = ez+nand gn(z) = ez+nfor all n∈N.Then fn∼gn
under φn(z) = z+n, n ∈N.Both the families have domain of normality as C.
Using the fact that the SD of a Mobius transformation is 0, one observes if Fand Gare
two families of meromorphic functions conjugate under the family Aof Mobius transforma-
tions, then their corresponding SD family satisfies the important relation (Sg◦φ)(φ′)2=Sf
for all f∈F, g ∈Gand φ∈Awhere f∼gunder φ.
References
[1] L. V. Ahlfors, Complex Analysis, McGraw-Hill, (1979).
NORMALITY AND SCHWARZIAN DERIVATIVES 5
[2] J. Grahl and S. Nevo, A note on spherical derivatives and normal families, arXiv:
1010.4654v1[math.CV], 12p. (2010).
[3] O. Lehto, Univalent Functions and Teichmuller Spaces, Springer-Verlag 109, (1987).
[4] J. Schiff, Normal families, Springer, New York, (1993).
[5] N. Steinmetz, Normal families and linear differential equations, arXiv: 1102.3104v1[math.CV], 3p.
(2011).
[6] L. Zalcman, Normal Families: New Perspectives, Bull. Amer. Math. Soc. 35 (1998), 215-230.
Department of Mathematics, University of Delhi, Delhi–110 007, India
E-mail address:dinukumar680@gmail.com
Department of Mathematics, Deen Dayal Upadhyaya College, University of Delhi,
Delhi–110 007, India
E-mail address:sanjpant@gmail.com
