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
Related Articles
- EP ID EP355313
- DOI 10.15330/ms.49.1.29-51
- Views 66
- Downloads 0
The analogue of Bernstein’s inverse theorem for the one class of the space of sequences (in Ukrainian)
We introduce the space of numerical sequences lp={x=(ξk)∞k=1:|x|=∑∞k=1|ξk|pk≤+∞} with a quasi-norm |⋅| for an every sequence p=(pk)∞k=1 of numbers pk from the interval (0,1] and we prove the analogue of the inverse of Be...
Convergence of some branched continued fractions with independent variables
In this paper, we investigate a convergence of associated multidimensional fractions and multidimensional \emph{J}-fractions with independent variables that are closely related to each other; the coefficients of its part...
Meromorphic function sharing three sets with its shift, q-shift, q-difference and analogous operators
Using the notion of weighted sharing of three sets we deal with the uniqueness problem of meromorphic function with its shift, q-shift, q-difference and analogous operator. A handful number of examples have been provided...
Fast growing meromorphic solutions of the systems of linear differential equations
Systems of linear differential equations that allow for dimension decrease are considered. Growth estimates for meromorphic vector-solutions are obtained. An essentially new feature is that there are no additional constr...
Pfluger-type theorem for functions of refined regular growth
Without a priori assumptions on zero distribution we prove that if an entire function f of noninteger order ρ has an asymptotic of the form log|f(reiθ)|=rρhf(θ)+O(rρδ(r)), E∌reiθ→∞, where h is the indicator of f, δ is a...