Estimates for Holomorphic Functions with Values in 0 , 1

Extension of classical Mandelbrojt’s criterion for normality to several complex variables is given. Some inequalities for holomorphic functions which omit values 0 and 1 are obtained.


Introduction
In 1929, Mandelbrojt [1] has asserted his criterion for normality of a family of holomorphic zero-free functions of one complex variables.
In [2], the author has proved a generalization of Mandelbrojt's criterion to several complex variables.In order to state this criterion precisely, we introduce some notations.
Let be a family of zero-free holomorphic functions in a domain and be a subdomain in such that , if 1 for all ; sup ln , if 1 for some , It is well known that a family of functions holo-morphic on a domain   all of which omits the values 0 and 1 is normal, so by the Theorem for some 0 and all r .f   But for this case we may obtain a more plain inequalities: Proposition 2. Let X K be the Kobayashi distance on a connected complex space Let be the family of all holomorphic functions on .

X
 X with values in where In the proof of this proposition, we combine the result of Lai [3] with the definition of the Kobayashi metric and obtain a very elementary proof of Proposition 3 in [4].
By hypothesis is normal, and therefore, the following two cases exhaust all the possibilities for sequence   : In case a) (respectively in case c)) we have and all sufficiently large.Hence Pluriharmonic functions form a subclass of the class of harmonic functions in (obviously proper for ).So by Harnack's inequality there exists some constant is bounded, which is a contradiction to (4).
Fix a point in 0 z  and define the families and by In case b), we have for all and hence finishes the proof in caseb).If   , then 1 f is holomorphic on  because f never vanishes.Also 1 f never vanishes and Hence reasoning similar to that in the above proof shows that and an analytic func-  .In fact
Let  denote the Poincaré distance on i.e., the di e f i  , stanc unction defined by the Poincaré metr c  ot take t (5) we the following inequality does n he values 0 and 1, from derive be a pair of points in X .Since X K , x y is an inner p ometric (see [6]), for ea 0 seud ch so the second inequality in (1) is proved.Since x and u y play symmetric roles, it is evident that the first ineq ality in (1) also holds.
For Remark 2. Proposition 2 holds also for holom nctions defined on an infinite dimensional complex Banach manifold with values in   0,1   , the same proof works.So we give here mo le proof of Proposition 3 in [4].


to Since is a family of zero-free holomorphic functions in a domain by Hurwit's theorem   f is either nowhere zero or identically equal to zero.Therefore the following three cases exhaust all the possibilities for sequence   : to a holomorphic function f which is zero-free on  Since it follows readily from (3) that to a holomorphic function or to  .The following two cases exhaust all the possibilities: a) There exists a subsequence k by Montel's theorem and hence we are done in casea).