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.
Let $A\in(-\infty,+\infty]$, $\Phi$ be a continuous function on $[a,A)$ such that for every $x\in\mathbb{R}$ we have$x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma)\colon \sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, $F(s)=\sum a_ne^{s\lambda_n}$ be a Dirichlet series absolutely convergent in the half-plane $\operatorname{Re}s<A$, $M(\sigma,F)=\sup\{|F(s)|\colon \operatorname{Re}s=\sigma\}$ and $G(\sigma,F)=\sum |a_n|e^{\sigma\lambda_n}$ for each $\sigma<A$. It is proved that if $\ln G(\sigma,F)\le(1+o(1))\Phi(\sigma)$, $\sigma\uparrow A$, then the inequality$$\varlimsup_{\sigma\uparrow A}\frac{M(\sigma,F')}{M(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1$$holds, and this inequality is sharp. % Abstract (in English)
Let $A\in(-\infty,+\infty]$, $\Phi:[a,A)\to\mathbb{R}$ be a continuous function such that $x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$ for every $x\in\mathbb{R}$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma):\sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ and $\Gamma(x)=(\widetilde{\Phi}(x)-\ln x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, and $F(s)=\sum\limits\limits_{n=0}^\infty a_ne^{s\lambda_n}$ be a Dirichlet series such that its maximal term $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge0\}$ and central index $\nu(\sigma,F)=\max\{n\ge0:|a_n|e^{\sigma\lambda_n}=\mu(\sigma,F)\}$ are defined for all $\sigma<A$. It is proved that if $\ln\mu(\sigma,F)\le(1+o(1))\Phi(\sigma)$ as $\sigma\uparrow A$, then the inequalities $$ \varlimsup_{\sigma\uparrow A}\frac{\mu(\sigma,F')}{\mu(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1,\qquad \varlimsup_{\sigma\uparrow A}\frac{\lambda_{\nu(\sigma,F')}}{\Gamma^{-1}(\sigma)}\le1, $$ hold, and these inequalities are sharp.
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