Geodesic
On the growth of the maximum modulus of an entire function depending on the growth of its central index
Ufa mathematical journal, Tome 3 (2011) no. 1, pp. 92-100

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Let $h$ be a positive function continuous on $(0,+\infty)$, $f(z)=\sum_{n=0}^\infty a_nz^n$ be an entire function, and $M_f(r)=\max\{|f(z)|\colon|z|=r\}$, $\mu_f(r)=\max\{|a_n|r^n\colon n\ge0\}$, and $\nu_f(r)=\max\{n\ge0\colon|a_n|r^n=\mu_f(r)\}$ be the maximum modulus, the maximal term, and the central index of the function $f$, respectively. We establish necessary and sufficient conditions for the growth of $\nu_f(r)$ under which $M_f(r)=O(\mu_f(r)h(\ln\mu_f(r)))$, $r\to+\infty$.
Keywords: entire function, maximum modulus, central index, order, lower order.
Mots-clés : maximal term 
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     author = {P. V. Filevych},
     title = {On the growth of the maximum modulus of an entire function depending on the growth of its central index},
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     url = {http://geodesic.mathdoc.fr/item/UFA_2011_3_1_a8/}
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P. V. Filevych. On the growth of the maximum modulus of an entire function depending on the growth of its central index. Ufa mathematical journal, Tome 3 (2011) no. 1, pp. 92-100. http://geodesic.mathdoc.fr/item/UFA_2011_3_1_a8/