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

Abstract

Let f be an analytic function in the disk D R = {z ∈ C : |z| < R}, R ∈ (0, +∞]. A point w ∈ D R is called 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 the zero set of f , i.e., d(w, f) = inf{|w − z| : f (z) = 0}. Let Φ be a continuous function on [a, ln R) such that xσ − Φ(σ) → −∞, σ ↑ ln R, for every x ∈ R. Let also Φ be the Youngconjugate function of Φ and Φ(x) = Φ(x)/x for all sufficiently large x. We prove that if ln M (r, f) ≤ (1 + o(1))Φ(ln r), r ↑ R, then lim |w|↑R d(w, f) Φ −1 (ln |w|) |w| ≥ C 0 , where C 0 = 0, 5416. .. . When the Taylor coefficients of f are nonnegative, the constant C 0 can be replaced by π, and the inequality obtained in this case is sharp.

Growth estimates for the maximal term and central exponent of the derivative of a Dirichlet series

“…In the following two lemmas, which are proved in [2], ϕ and x 0 are defined by Φ in the same way as in Lemma 1.…”

“…Since for each Φ ∈ Ω A we have Φ(x)/x = Φ(x) → A as x → +∞, for a Dirichlet series of the form (1) whose existence follows from Theorem 4 we obtain β(F) = A by (2). If Φ ∈ Ω A and a Dirichlet series of the form (1) with β(F) = A satisfies (16), then, by Theorem 3 and the left of inequalities (10), for all σ ∈ [σ 2 , A) we obtain λ ν(σ,F) ≤ Φ −1 (σ).Since λ ν(σ,F) = (ln µ(σ, F)) ′ + for every σ < β(F), this fact is easy to prove without using Theorem 3 (see[2, Lemma 5] or[3, Lemma 4]).…”

Growth estimates for a Dirichlet series and its derivative

“…Hence, T Φ (F ) = T Φ (F ) and K(σ, F ) = (ln M (σ, F )) ′ , σ < A. Therefore, as is easy to prove (see Lemma 4 below; see also Lemma 1 in [2]), for such series, without any conditions on the sequence λ of its exponents and on a function Φ ∈ Ω A , the equality T Φ (F ) = 1 (or the identical equality T Φ (F ) = 1) implies the inequality…”

“…Proof of Theorem 1. Suppose that A ∈ (−∞, +∞], Φ ∈ Ω A , F ∈ D A is a Dirichlet series of the form (1) such that T Φ (F ) ≤ 1, and prove that inequality (5) holds. For all s ∈ C with Re z < A and each N ∈ R we put…”

“…For a Dirichlet series F ∈ D A of the form (1) and every σ < A we put Let Φ : D Φ → R be a real function. We say that Φ ∈ Ω A if the domain D Φ of Φ is an interval of the form [a, A), Φ is continuous on D Φ , and the following condition ∀x ∈ R : lim σ↑A (xσ − Φ(σ)) = −∞ (2) holds. It is easy to see that in the case A < +∞ condition (2) is equivalent to the condition Φ(σ) → +∞, σ → A − 0, and in the case A = +∞ this condition is equivalent to the condition Φ(σ)/σ → +∞, σ → +∞.…”