Wiman’s type inequality for multiple power series in an unbounded cylinder domain
Journal Title: Математичні Студії - Year 2018, Vol 49, Issue 1
Abstract
In this paper we prove some analogues of Wiman’s inequality for analytic f(z) and random analytic functions f(z,t) on T=Dl×Cp−l, l∈N, 1≤l≤p, I={1,…,l}, J={l+1,…,p} of the form f(z)=∑+∞∥n∥=0anzn, f(z,t)=∑+∞∥n∥=0anZn(t)zn respectively. Here Z=(Zn) is a multiplicative system of random variables on the Steinhaus probability space, uniformly bounded by the number 1. In particular, there are proved the following statements: For every δ>0 there exist sets E1=E1(δ,f), E2=E2(δ,f)⊂[0,1)l×(1,+∞)p−l of asymptotically finite logarithmic measure, such that the inequalities Mf(r)≤μf(r)∏i∈I1(1−ri)1+δlnp/2+δ(μf(r)∏i∈I11−ri)(∏j∈Jlnrj)p+δ,Mf(r,t)≤μf(r)∏i∈I1(1−ri)1/2+δlnp/4+δ(μf(r)∏i∈I11−ri)(∏j∈Jlnrj)p/2+δ. hold for all r∈T∖E1 and for all r∈T∖E2 a.s. in t, respectively. Also is proved sharpness of the obtained inequalities.
Authors and Affiliations
Andriy Kuryliak, Volodymyr Tsvigun
Keywords
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- EP ID EP355313
- DOI 10.15330/ms.49.1.29-51
- Views 71
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