NEW ZEALAND JOURNAL OF MATHEMATICS
Volume 23 (1994), 91-92
ON A N O RM ALITY CRITERION OF H.L. ROYDEN
W il h e l m S c h w ic k
(Received December 1993)
Abstract. The author proves a generalization of a normality criterion of H.L. Royden and gives an application to normality considerations for families of solutions of
differential equations.
In [Roy] Royden (compare [Sch], p. 76) is proved the following normality crite­
rion, which has recently been generalized by Hinkkanen [Hin].
Theorem. Let F be a family of meromorphic functions in a domain G c C with
the property that for each compact set K C G there is a positive increasing function
Hk such that
\f'{z)\ < hK {\f(z)\)
for all f E F and z E K . Then F is normal in G.
Obviously the result has its origin in the sufficiency of Marty’s criterion. We are
able to prove the theorem for a larger class of functions h x for which the estimate
holds. Let H be the set of all functions h :IRj —j►IRq" U {+ o o }, for which xq eIRq
exists such that h is bounded in a neighbourhood of x q . We prove:
Theorem. Let F be a family o f meromorphic functions in a domain G
C (D with
the property that for each compact set K C G there is a function h x E H such
that
\ f'( z ) \ < h K (\f(z)\)
for all f E F and z E K . Then F is normal in G.
Proof. Let z0 E G and K r(z 0)
C G.
Then exists ho E H with
\f'(z)\ < h0(\f(z)\)
for f E F and 2 E K r(zo ) and xo EIRq" such that ho(x ) < M for \x — xo| < S.
W.l.o.g. we may assume that xq / 0. If F is not normal at zo, it follows from
[Zal], Lemma 3 (compare [Sch], p. 152) that there exist zn —> z0, pn —>0 (pn > 0)
and f n E F with f n(zn + pnz) => g(z) locally uniformly on compact subsets of (D in
the spherical sense, where g is a nonconstant meromorphic function in the plane.
Choose z\ E (Dwith |<?(zi)| — xq and </(2 i) 7^ 0 . Then
Pufni^n “I- Pn-21) = (fn(Zn “t- Pn^)) (^l)
* 9 (^l) 7^ 0’
On the other hand we have f n{zn + pnz 1) —>g{zi) and therefore, if n is sufficiently
large,
Pn\fn(Zn
“t- Pn^l)| ^ Pn^O ( \ f n ( z n + Pn-^l))) 5: Pn ' M
> 0,
1991 A M S Mathematics Subject Classification: Primary 30D 45, Secondary 30D35.
92
WILHELM SCHWICK
which is a contradiction.
The result can easily be applied to normality considerations of solutions of differ­
ential equations. For example, the family of all solutions of the differential equation
w =
w + 1
which are defined in a common domain is normal, since
w+ 1
<
e H.
The theorem may be generalized to higher derivatives.
T h eorem . Let k be an integer with k > 2 and F be a family of meromorphic
functions in a domain G C C such that each f € F has only zeros with multiplicity
> k. If for each compact subset K C G there exists Hk € H with
l / (fc)M I < M l/M l)
for f € F and z € K , then F is normal in G.
The theorem can be proved as the above one. It is only necessary to ensure that
the limit function g is not a polynomial of degree k — 1, which follows from the
multiplicity condition for the zeros of / G F .
R eferen ces
[Hin] A. Hinkkanen, Normal families and Ahlfors’s five islands theorem, New Zeal.
J. Math. 22 (1993), 39-41.
[Roy] H.L. Royden, A criterion for the normality of a family of meromorphic func­
tionsi, Ann. Acad. Sci. Fenn. (Series A.I.) 10 (1985), 499-500.
[Sch] J.L. Schiff, Normal Families, Springer Verlag, New York, 1993.
[Zal] L. Zalcman, A heuristic principle in complex function theory, Amer. Math.
Monthly 82 (1975), 813-817.
W ilhelm Schwick
Fachbereich Mathematik
Universitat Dortmund
44221 Dortmund
GERMANY