Bernstein-type inequalities for analytic functions represented by power series

Abstract

Let $R\in (0,+\infty]$ and $\mathbb{D}_R=\{z\in \mathbb{C}\colon |z|<R\}$. Denote by
$\mathcal{A}_R$ the class of all functions $f$ analytic in $\mathbb{D}_R$ such that $f(z)\not\equiv0$. For any function $f\in\mathcal{A}_R$, let $M(r,f)=\max\{|f(z)|\colon |z|=r\}$ be the maximum modulus, $K (r,f)=rM(r,f')/M(r,f)$, and $\mu(r,f)=\max\{|a_n(f)|r^n\colon n\ge 0 \}$ be the maximal term of the Maclaurin series of the function $f$, where $a_n(f)$ denotes the $n$-th coefficient of this series. Suppose that $\Phi$ is a continuous function on $[a,\ln R)$ such that for every $x\in\mathbb{R}$ we have $x\sigma-\Phi(\sigma )\to-\infty$ as $\sigma\uparrow \ln R$, and let $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma)\colon \sigma\in D_\Phi\}$ be the Young conjugate function of $\Phi$, $\varphi(x)=\widetilde{\Phi}'_+(x)$ for all $x\in\mathbb{ R}$, and $\Gamma(x)=(\widetilde{\Phi}(x)-\ln x)/x$ for all sufficiently large $x$. Put

$\displaystyle
\Delta=\varlimsup_{x\to+\infty}\frac{\ln x}{\Phi(\varphi(x))},\
t(f)=\varlimsup_{r\uparrow R}\frac{\ln\mu(r,f)}{\Phi(\ln r)},\
k(f)=\varlimsup_{r\uparrow R} \frac{K(r,f)}{\overline{\Phi}\,\!^{-1}(\ln r)},\
k_{1}(f)=\varlimsup_{r\uparrow R} \frac{K(r,f)}{\Gamma^{-1}(\ln r)},
$

where $f\in\mathcal{A}_R$. We prove the following results:

(a) for any function $f\in\mathcal{A}_{R}$ such that $t(f)\le1$, the inequality $k_{1}(f)\le 1$ holds;
(b) for an arbitrary positive sequence $(r_n)$ increasing to $R$, there exists a function $f\in\mathcal{A}_{R}$ such that $t(f)=1$ and $\varlimsup\limits_{n \to +\infty}K(r_n,f)/\Gamma^{-1}(\ln r_n)=1$;
(c) for any function $f\in\mathcal{A}_{R}$ such that $t(f)\le1$, the inequality $k(f)\le 1+\Delta$ holds;
(d) there exists a function $f\in\mathcal{A}_{R}$ such that $t(f)=1$ and $k(f)=1+\Delta$.