Maximum modulus of entire functions of two variables and arguments of coefficients of double power series
Let $mathcal{L}$ be {the} class of positive continuousfunctions on $(-infty,+infty)$ and {let} $mathcal{L}_+^2$ be{the} class of positive continuous increa-sing with respect toeach variable functions $gamma$ in $mathbb{R}^2$ such that $gamma(r_1,r_2)o +infty $ as $r_1+r_2o+infty.$ {We provethe follo...
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Ivan Franko National University of Lviv
2011-11-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/texts/2011/36_2/162-175.pdf |
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| author | O. B. Skaskiv A. O. Kuryliak |
| author_facet | O. B. Skaskiv A. O. Kuryliak |
| author_sort | O. B. Skaskiv |
| collection | DOAJ |
| description | Let $mathcal{L}$ be {the} class of positive continuousfunctions on $(-infty,+infty)$ and {let} $mathcal{L}_+^2$ be{the} class of positive continuous increa-sing with respect toeach variable functions $gamma$ in $mathbb{R}^2$ such that $gamma(r_1,r_2)o +infty $ as $r_1+r_2o+infty.$ {We provethe following} statement: {for all} entire functions of theform $ f(z_1,z_2)=sum_{n+m=0}^{+infty}a_{nm}z_1^nz_2^m$ suchthat $ |a_{nm}|leqexp{-(n+m)psi(n,m)} mbox{ for } n+mgeqk_0(f)$ and functions $f(z_1,1), f(1,z_2)$ are transcendent, $psiinmathcal{L}_+^2, $ {the} inequality$$mathfrak{M}_f(r_1,r_2) =O(M_f(r_1,r_2) h(ln M_f(r_1,r_2))),hinmathcal{L}, r^{vee}=min{r_1,r_2}o+infty,$${holds} where $M_f(r_1,r_2)=max{|f(z_1,z_2)|colon|z_1|=r_1,|z_2|=r_2},$$mathfrak{M}_f(r_1,r_2)=sum_{n+m=0}^{+infty}|a_{nm}|imes$$imes r_1^nr_2^m,$ {if and only if}egin{equation*}(finl)colon\sqrt{r_1r_2}=Oig(h(gamma(r_1,r_2)psi(r_1,r_2))ig), ^{vee}o+infty.end{equation*} |
| format | Article |
| id | doaj-art-fe59724091484601a5935ceada59d80a |
| institution | DOAJ |
| issn | 1027-4634 |
| language | deu |
| publishDate | 2011-11-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-fe59724091484601a5935ceada59d80a2025-08-20T03:22:26ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342011-11-01362162175Maximum modulus of entire functions of two variables and arguments of coefficients of double power seriesO. B. SkaskivA. O. KuryliakLet $mathcal{L}$ be {the} class of positive continuousfunctions on $(-infty,+infty)$ and {let} $mathcal{L}_+^2$ be{the} class of positive continuous increa-sing with respect toeach variable functions $gamma$ in $mathbb{R}^2$ such that $gamma(r_1,r_2)o +infty $ as $r_1+r_2o+infty.$ {We provethe following} statement: {for all} entire functions of theform $ f(z_1,z_2)=sum_{n+m=0}^{+infty}a_{nm}z_1^nz_2^m$ suchthat $ |a_{nm}|leqexp{-(n+m)psi(n,m)} mbox{ for } n+mgeqk_0(f)$ and functions $f(z_1,1), f(1,z_2)$ are transcendent, $psiinmathcal{L}_+^2, $ {the} inequality$$mathfrak{M}_f(r_1,r_2) =O(M_f(r_1,r_2) h(ln M_f(r_1,r_2))),hinmathcal{L}, r^{vee}=min{r_1,r_2}o+infty,$${holds} where $M_f(r_1,r_2)=max{|f(z_1,z_2)|colon|z_1|=r_1,|z_2|=r_2},$$mathfrak{M}_f(r_1,r_2)=sum_{n+m=0}^{+infty}|a_{nm}|imes$$imes r_1^nr_2^m,$ {if and only if}egin{equation*}(finl)colon\sqrt{r_1r_2}=Oig(h(gamma(r_1,r_2)psi(r_1,r_2))ig), ^{vee}o+infty.end{equation*}http://matstud.org.ua/texts/2011/36_2/162-175.pdfentire functionspower seriesmaximum modulus |
| spellingShingle | O. B. Skaskiv A. O. Kuryliak Maximum modulus of entire functions of two variables and arguments of coefficients of double power series Математичні Студії entire functions power series maximum modulus |
| title | Maximum modulus of entire functions of two variables and arguments of coefficients of double power series |
| title_full | Maximum modulus of entire functions of two variables and arguments of coefficients of double power series |
| title_fullStr | Maximum modulus of entire functions of two variables and arguments of coefficients of double power series |
| title_full_unstemmed | Maximum modulus of entire functions of two variables and arguments of coefficients of double power series |
| title_short | Maximum modulus of entire functions of two variables and arguments of coefficients of double power series |
| title_sort | maximum modulus of entire functions of two variables and arguments of coefficients of double power series |
| topic | entire functions power series maximum modulus |
| url | http://matstud.org.ua/texts/2011/36_2/162-175.pdf |
| work_keys_str_mv | AT obskaskiv maximummodulusofentirefunctionsoftwovariablesandargumentsofcoefficientsofdoublepowerseries AT aokuryliak maximummodulusofentirefunctionsoftwovariablesandargumentsofcoefficientsofdoublepowerseries |