|
|
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 L be the class of positive continuous functions on
(−∞,+∞) and let L2+ be the class of positive
continuous increasing with respect to each variable functions
γ in R2 such that γ(r1,r2)→+∞ as r1+r2→+∞. We prove the following} statement:
for all entire functions of the form f(z1,z2)=∑+∞n+m=0anmzn1zm2 such that |anm|≤exp{−(n+m)ψ(n,m)} for n+m≥k0(f) and functions f(z1,1),f(1,z2) are transcendent,
ψ∈L2+, the inequality
Mf(r1,r2)=O(Mf(r1,r2)h(lnMf(r1,r2))), h∈L, r∨=min
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 |
| 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 |
| Full text of paper |
PDF |
| Table of content of issue |
HTML |