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Maximum modulus of entire functions of two variables and
arguments of coefficients of double power series |
| Author |
O. B. Skaskiv, A. O. Kuryliak
matstud@franko.lviv.ua,
kurylyak88@gmail.com
Lviv Ivan Franko National University
|
| Abstract |
Let $\mathcal{L}$ be the class of positive continuous functions on
$(-\infty,+\infty)$ and let $\mathcal{L}_+^2$ be the class of positive
continuous increasing with respect to each variable functions
$\gamma$ in $\mathbb{R}^2$ such that $ \gamma(r_1,r_2)\to +\infty
$ as $r_1+r_2\to+\infty.$ We prove the following} statement:
for all entire functions of the form $
f(z_1,z_2)=\sum_{n+m=0}^{+\infty}a_{nm}z_1^nz_2^m$ such that $
|a_{nm}|\leq\exp\{-(n+m)\psi(n,m)\} \mbox{ for } n+m\geq k_0(f)$ and functions $f(z_1,1), f(1,z_2)$ are transcendent,
$ \psi\in\mathcal{L}_+^2, $ the inequality
$$
\mathfrak{M}_f(r_1,r_2)=O(M_f(r_1,r_2) h(\ln M_f(r_1,r_2))),\
h\in\mathcal{L},\
r^{\vee}=\min\{r_1,r_2\}\to+\infty,
$$
holds where $M_f(r_1,r_2)=\max\{|f(z_1,z_2)|\colon
|z_1|=r_1,|z_2|=r_2\},$\
$\mathfrak{M}_f(r_1,r_2)=\sum_{n+m=0}^{+\infty}|a_{nm}|\times$ $\times r_1^nr_2^m,$
if and only if
\begin{equation*}
(\forall\gamma\in\mathcal{L}_+^2)\colon\ \sqrt{r_1r_2}=O\big(h(\gamma(r_1,r_2)\psi(r_1,r_2))\big), \
r^{\vee}\to+\infty.
\end{equation*} |
| Keywords |
entire functions; power series; maximum modulus |
| DOI |
doi:10.30970/ms.36.2.162-175
|
| Reference |
1. Brinkmeier H. Uber das Mass der Bestimmtheit des Wachstum einer ganzen transzendenten Funktion
durch die absoluten Betrage der Koeffizienten ihrer Potenzreihe// Math. Ann. - 1926. - B.96, №1. -
S. 108-118.
2. Zelisko M.M., Sheremeta M.M. On influence of coefficients arguments for Dirichlet series on its growth//
Mat. Stud. - 2006. - V.26, №1. - P. 81-85. (in Ukrainian)
3. Filevych P.V. Wiman-Valiron type inequalities for entire and random entire functions of finite logarithmic
order// Sib. Mat. Zh. - 2001. - V.42, №3. - P. 683-692. (in Russian) English transl. in Sib. Math. J. 2001,
V.42, №3, 579-586.
4. Filevych P.V. On influence of the arguments of coefficients of a power series expansion of an entire
function on the growth of the maximum of its modulus// Sib. Mat. Zh. - 2003. - V.44, №3. - P. 674-685.
(in Russian) English transl. in Sib. Math. J. 2003, V.44, ‡‚3, 529.538.
5. Kahane J.-P. SЃLeries de Fourier absolument convergentes. - Ergebnisse der Mathematik und ihrer
Grenzgebiete. Bd. 50. - Berlin-Heidelberg-New York: Springer Verlag, 1970.
|
| Pages |
162-175 |
| Volume |
36 |
| Issue |
2 |
| Year |
2011 |
Journal |
Matematychni Studii |
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