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This article is cited in 6 scientific papers (total in 6 papers)
Minimum modulus of lacunary power series and $h$-measure of exceptional sets
T. M. Saloa, O. B. Skaskivb a Institute of Applied Mathematics and Fundamental Sciences,
National University "Lvivs’ka Polytehnika",
Stepan Bandera str. 12,
79013, Lviv, Ukraine
b Department of Mechanics and Mathematics,
Ivan Franko National University of L’viv,
Universytetska str. 1,
79000, Lviv, Ukraine
Abstract:
We consider some generalizations of Fenton theorem for the entire functions represented by lacunary power series.
Let $f(z)=\sum_{k=0}^{+\infty}f_kz^{n_k}$, where $(n_k)$ is a strictly
increasing sequence of non-negative integers. We denote by
\begin{align*}
&M_f(r)=\max\{|f(z)|\colon |z|=r\},
\\
&m_f(r)=\min\{|f(z)|\colon |z|=r\},
\\
&
\mu_f(r)=\max\{|f_k|r^{n_k}\colon k\geq 0\}
\end{align*}
the maximum modulus,
the minimum modulus and the maximum term of $f,$ respectively.
Let $h(r)$ be a positive continuous function
increasing to infinity on $[1,+\infty)$ with a non-decreasing
derivative. For a measurable set $E\subset [1,+\infty)$ we introduce
$h-\mathrm{meas}\,(E)=\int_{E}\frac{dh(r)}{r}.$
In this paper we establish
conditions guaranteeing that the relations
$$
M_f(r)=(1+o(1)) m_f(r),\quad M_f(r)=(1+o(1))\mu_f(r)
$$
are true as $r\to+\infty$ outside some exceptional set $E$ such that $h-\mathrm{meas}\,(E)<+\infty$. For some subclasses we obtain necessary and sufficient conditions. We also provide similar
results for entire Dirichlet series.
Keywords:
lacunary power series, minimum modulus, maximum modulus, maximal term, entire Dirichlet series, exceptional set, $h$-measure.
Received: 22.07.2016
Citation:
T. M. Salo, O. B. Skaskiv, “Minimum modulus of lacunary power series and $h$-measure of exceptional sets”, Ufa Math. J., 9:4 (2017), 135–144
Linking options:
https://www.mathnet.ru/eng/ufa410https://doi.org/10.13108/2017-9-4-135 https://www.mathnet.ru/eng/ufa/v9/i4/p137
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