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
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- DOI 10.30970/ms.52.1.10-23
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