 Advances in Pure Mathematics, 2013, 3, 586-589 http://dx.doi.org/10.4236/apm.2013.36075 Published Online September 2013 (http://www.scirp.org/journal/apm) Estimates for Holomorphic Functions with Values in 0,1 P. V. Dovbush Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Kishinev, Republic of Moldova Email: peter.dovbush@gmail.com Received June 29, 2013; revised July 28, 2013, accepted August 24, 2013 Copyright © 2013 P. V. Dovbush. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract 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. Keywords: Complex Space; Holomorphic Functions 1. Introduction In 1929, Mandelbrojt  has asserted his criterion for normality of a family of holomorphic zero-free functions of one complex variables. In , the author has proved a generalization of Man- delbrojt’s criterion to several complex variables. In order to state this criterion precisely, we introduce some nota- tions. Let be a family of zero-free holomorphic func- tions in a domain and be a subdomain in such that nDD. So that the quantities  ,,ln ,if1 for all ;sup ln, if1 for some ,zw DmfDfz fww DfwfwwD  ,,,supzw DfzmfDfw ,min ,,,LfDmfDm fD, are well defined for each function . fTheorem 1. (See .) Let be a family of holo- morphic functions in a domain with values in Then is normal in if and only if for each point 0 there exists a ball 0, 1z00,Bz r such that the the set of quantities 00,,LfBzr , is bounded. ,f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 00,,LfBzr for some 0 and all r.f But for this case we may obtain a more plain inequalities: Proposition 2. Let XK be the Kobayashi distance on a connected complex space Let be the family of all holomorphic functions on .XX with values in 0,1. Then, for all ,xyX and all ,flogexp ,logexp, ,XXcfxKxy cfyKxy (1) where 2144.3768796 .4c Furthermore, if there exists continuous logf on X then logexp ,logexp,.XXcfxKxycfyKxy (2) In the proof of this proposition, we combine the result of Lai  with the definition of the Kobayashi metric and obtain a very elementary proof of Proposition 3 in . 2. The Proof of Mandelbrojt’s Criterion Proof of Theorem 1.  Fix a point in  and con- 0zCopyright © 2013 SciRes. APM P. V. DOVBUSH 587sider a ball 0,Bz rLfB. Suppose that  is normal in but the set for some 00,,zr,f,0,rr is unbounded. Then there exists a sequence jf suchthat 00 , ,LfBzr forall .jj  (3) By hypothesis is normal, and therefore, the fol-lowing two cases exhaust all the possibilities for se- quence :jf 1) The sequence jf has a subsequence kjf which converges uniformly on 00,Bzr to a holomor- phic function ;f 2) The sequence jf has a subsequence kjf which converges uniformly on 00,Bzr to Since is a family of zero-free holomorphic functions in a do- main 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 :jf a) The sequence jf has a subsequence kjf which converges uniformly on 00,Bzr to the holo- morphic function 0;fb) The sequence jf has a subsequence kjf which converges uniformly on 00,Bzr to a holomor- phic function f which is zero-free on 00,Bzr; c) The sequence jf has a subsequence kjf which converges uniformly on 00,Bzr to .Since it follows readily from (3) that 00 ,, jBzr for all Lf j (4) In case a) (respectively in case c)) we have 12kjfz (respectively 2kjfz for all 00,zBzr and all sufficiently large. Hence kln kjfz is a negative (respectively positive) pluri- harmonic function in Pluriharmonic functions form a subclass of the class of harmonic functions in (obviously proper for ). So by Harnack’s inequality there exists some constant 0Bz,r0,Bz r1n00 0,,, ,CCBzrBzr 1,,C that 00,Bln for all and ,lnkkjjfz Czwzrfw and hence 00,,kjmf BzrC for all suffi- ciently large. kIn case b), we have lim kjkfzfz  for all 0,.zBzr It follows 00lim uniformly for and ,kkjkjfz fz zwBzrfwfw  The function fzfw is holomorphic on 00 00,Bz rBz r,, it follows that 00,,mfBzr is bounded. Since 00,,kjLf Bzr is the minimum of 00,,kjmf Bzr and 00,,kjmf Bzrthe set of quantities 00,,,Lf Bzrk,kj is bounded, which is a contradiction to (4). Fix a point in 0z and define the families and by 0,1ffz  , 0,1ffz. It will be shown that is normal in Ω and that is normal in ,.CΩC To prove that the family 0,1ffz is normal, it is sufficient to show that each sequence jf contains a subsequence converging locally uniformly in 00,Bzr to a holomorphic function or to . The following two cases exhaust all the possibilities: a) There exists a subsequence kjf such that for any k the function ln kj does not vanish in 00,;Bz r fb) For each j there exists such that 00,zrjzBln 0.jjfz In case a), we have that 1kj in for all elements of the sequence. Such a subsequence is normal in 00,Bzrf00,Bzr by Montel’s theorem and hence we are done in casea). In case b), we have for all 00,,jmf Bzr.j Therefore, according to the hypothesis, 00,,jmfBzr C for all and some constant j0.C It follows that jfC in for all 00,Bzr,j which means thatjf is a normal family in 00,Bzrf and hence finishes the proof in caseb). If , then 1f is holomorphic on  because f never vanishes. Also1f never vanishes and 01fz 1. Hence reasoning similar to that in the above proof shows that 1:ff is also normal in 00,Bz r. So if jf is a sequence in  there is a subsequence kjf and an analytic func- tion on h00,Bzr such that1kjf converges in 00,Bzr to . By the generalized Hurwitz’s Theorem, either hh0 or never vanishes. If h0h it is easy to see that kj uniformly on compact subsets of f00.,rBz If never vanishes then h1h is Copyright © 2013 SciRes. APM P. V. DOVBUSH 588 analytic and it follows that 1kjfzhz uniformly on compact subsets of . 00,BzrIt follows that and are normal at 0 so that the union is normal in z00,rBz Since normality is a local property, is a normal family in  This completes the proof of the theorem. Remark 1. It should be pointed out that the above theorem is not true if the condition “for each point there exists a ball 0Ωz00,r,, .Bz00r such that the the set of quantities ,LfBz ,fF ised” is replaced by the condition “the corresponding family of boundfunctions given by ,fzwfzf w is locally bounded on ΩΩ (cf. [5, Theorem 2.2.8]). , consTo see thisider the family :j1jFz of holo- morphic functions. If we take ::121,Az z then AF is a set of bounded (b- morphunctions in y 1) zero-free holoic fA so Montel’s theorem guaran- tees that F is normal. It is plain by inspection that the family 1jjjis not locally bounded on ,zwAA while 1ln lnjjjzw is a locally bounded family on .AA.2.8 in  is not true. Hence Theorem 23. Estimates for Holomorphic Functions Pr m of Landau Which Omit the Values 0 and 1 oof of Proposition 2. The classical theoremay be stated in the form that if the function fz is holomorphic in the unit disk Δ:1zz nd  1, then adoes not take the values 0 and 0f has a bound depending only on 0f. In fact 0f has a bound depending only on 0f   0 log0cf (5) where 02ff2144.37687964c (see, for example, ). Let  denote the Poincaré distance on i.e., the die fi, stancunction defined by the Poincaré metr c 2224.1dzdzds z For define Δz1zwwz zw . Since zf ot take t (5) we the following inequality does nhe values 0 and 1, fromderive 020log0.zzzzffcf  (6) Let 210zfbe a pair of points in X. Since XK,xy is an inner pometric (see ), for ea 0seud ch t exist an integer 1,l here1,, Δ,,lHol Xand 1,,a 0, 1la satng isfyi10,y 10j , forjja1, ,jl1 and ,laxl and 10,, 2.lxyjXjaK Set .jjgf From inequality (6), we obtain  22 for 0,.1logjjjjgt tatgtc gt (7) Since for 0,jta   loglog logjjjjjgt gtgt tgtgtt tfrom (7), we obtain 22log .1jgttt (8) If we integrate both sides fromsulogc 0t to , the re- jalt becomes loglog0, .log 0jjjjcga acg Then  loglog, .log XcfxKxycfy , we finally get Letting 0 log fxexp ,log XcKxycfy so the second inequality in (1) is proved. Since x and uy play symmetric roles, it is evident that the first ine- qality in (1) also holds. For obtaining inequalities (2), let us notice that there exists continuous log logjg on 0,jta. Since  log logloglogjjgtgtttwe have  Copyright © 2013 SciRes. APM P. V. DOVBUSH Copyright © 2013 SciRes. APM 589References   log loglog .jjjjgtgtggt ttcgtt tFrom this inequality and inequality (8), we obtain  S. Mandelbrojt, “Sur les Suites de Fonctions Holomorphes. Les Suites Correspondantes des Fonctions Dérivées. Fonctions Entières,” Journal de Mathématiques Pures et Appliquées, Vol. 9, No. 8, 1929, pp. 173-196. http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1929_9_8_A10_0 21jjttIntegrating both sides of this inequality as abovobofuor Robert B. Burckel for all 2log log for 0,.cgt ta  P. V. Dovbush, “On a Normality Criterion of S. Man- delbrojt,” 2013. http://arxiv.org/abs/1302.1695 e we  W. T. Lai, “The Exact Value of Heyman’s Constant in Landau’s Theorem,”Scientia Sinica, Vol. 22, 1979, pp. 129-133. tain the inequality (2). The proof of the theorem is now complete.  rphic  E. M. Chirka, “Harnack Inequalities, Kobayashi Distances and Holomorphic Motions,” Proceedings of the Steklov Institute of Mathematics, Vol. 279, No. 1, 2012, pp. 194- 206. Remark 2. Proposition 2 holds also for holomnctions defined on an infinite dimensional complex Banach manifold with values in 0, 1, the same proof works. So we give here mole proof of Proposition 3 in . 4. Acknowledgements re simpdoi:10.1134/S0081543812080135  J. L. Shiff, “Normal Families,” Springer-Verlag, New York, 1993. doi:10.1007/978-1-4612-0907-2  S. Kobayashi, “Hyperbolic Complex Space,” Springer- Verlag, New York, 1998. doi:10.1007/978-3-662-03582-5 I would like to thank Profess his help during my work on this paper.