Wiman’s Type Inequality in Multiple-Circular Domain

Andriy Kuryliak,Oleh Skaskiv

doi:10.3390/axioms10040348

Abstract

In the paper we prove for the first time an analogue of the Wiman inequality in the class of analytic functions f∈A0p(G) in an arbitrary complete Reinhard domain G⊂Cp, p∈N represented by the power series of the form f(z)=f(z1,⋯,zp)=∑‖n‖=0+∞anzn with the domain of convergence G. We have proven the following statement: If f∈Ap(G) and h∈Hp, then for a given ε=(ε1,…,εp)∈R+p and arbitrary δ>0 there exists a set E⊂|G| such that ∫E∩Δεh(r)dr1⋯drpr1⋯rp<+∞ and for all r∈Δε∖E we have Mf(r)≤μf(r)(h(r))p+12lnp2+δh(r)lnp2+δ{μf(r)h(r)}∏j=1p(lnerjεj)p−12+δ. Note, that this assertion at p=1,G=C,h(r)≡const implies the classical Wiman–Valiron theorem for entire functions and at p=1, the G=D:={z∈C:|z|<1},h(r)≡1/(1−r) theorem about the Kővari-type inequality for analytic functions in the unit disc D; p>1 implies some Wiman’s type inequalities for analytic functions of several variables in Cn×Dk, n,k∈Z+,n+k∈N.

Highlights

Notations and Preliminaries2021, 10, 348. https://doi.org/Let C, R, Z, N be sets of complex numbers, real numbers, integers, and positive p integers, respectively, and Z+ = N ∪ {0}
We denote by A0 (G), p ∈ N, the class of an analytic functions f in a complete Reinhardt domain G ⊂ C p, represented by the power series of the form
We say that a domain G ⊂ C p is the complete Reinhardt domain if: ( a) z = (z1, . . . , z p ) ∈ G =⇒ (∀ R = ( R1, . . . , R p ) ∈ [0, 1] p ) : Rz = ( R1 z1, . . . , R p z p ) ∈ G

Summary

Introduction

We denote by A0 (G), p ∈ N, the class of an analytic functions f in a complete Reinhardt domain G ⊂ C p , represented by the power series of the form. Reinhardt domain G with a center at z = 0 can be represented in G by the series of form (1). The domain of convergence of each series of form (1) is the logarithmically-convex complete Reinhardt domain with the center z = 0. We say that a domain G ⊂ C p is the complete Reinhardt domain if:. The Reinhardt domain G is called logarithmically-convex if the image of the set Reinhardt domain is a disc. 1) ∈ R p , B p (r ), Π p (r ) (r > 0) are the logarithmically-convex complete Reinhardt domains.

Wiman’s Type Inequality for Analytic Functions of One Variable

Wiman’s Type Inequality for Analytic Functions of Several Variables

Main Result

Auxiliary Lemmas

Proof of the Main Theorem

Discussion