Distance between a maximum modulus point and zero set of an analytic function

Apply

Distance between a maximum modulus point and zero set of an analytic function

Journal Title: Математичні Студії - Year 2019, Vol 52, Issue 1

Abstract

Let f be an analytic function in the disk DR={z∈C:|z|≤R}, R∈(0,+∞]. We call a point w∈DR a maximum modulus point of f if |f(w)|=M(|w|,f), where M(r,f)=max{|f(z)|:|z|=r}. Denote by d(w,f) the distance between a maximum modulus point w and zero set of f, i.e., d(w,f)=inf{|w−z|:f(z)=0}. Let Φ be a continuous function on [a,lnR) such that xσ−Φ(σ)→−∞, σ↑lnR, for every x∈R. Let also Φ˜ be the Young-conjugate function of Φ and Φ¯¯¯¯(x)=Φ˜(x)/x for all sufficiently large x. We prove that if lnM(r,f)≤(1+o(1))Φ(lnr),r↑R, then lim−−−|w|↑Rd(w,f)Φ¯¯¯¯−1(ln|w|)|w|≥C0, where C0=0,5416…. When the Taylor coefficients of f are nonnegative, the constant C0 can be replaced by π, and the inequality obtained in this case is sharp.

Authors and Affiliations

S. I. Fedynyak, P. V. Filevych

Keywords

analytic function; maximum modulus; maximum modulus point; zero set

Infinite-modal approximate solutions of the Bryan-Pidduck equation

Abstract The nonlinear integro-differential Bryan-Pidduck equation for a model of rough spheres is considered. An approximate solution is constructed in the form of an infinite linear combination of some Maxwellian mode...

Analysis the solutions of the differential nonlinear equations describing the information spreading process with jump discontinuity

In this paper, we introduce a mathematical model of spreading any type of information. The model has the form of a system of nonlinear differential equations with non-stationary parameters. We have suggested the explicit...

EP ID EP673792
DOI 10.30970/ms.52.1.10-23
Views 52
Downloads 0