On growth order of solutions of differential equations in a neighborhood of a branch point
Let $M_k$ be {the} set of $k$-valued meromorphic in$G={zcolon r_0leqslant |z|}$ functions with {a}~branch point of order$k-1$ {at} $infty$; let $E_ast$ be a set of circles {with finite} sum of radii. Denote$M_ast(r,f)=max|f(z)|, zin{te^{iheta}colon 0leqslanthetaleqslant2kpi,_0leqslant tleqslant r}s...
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Ivan Franko National University of Lviv
2013-10-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/texts/2013/40_1/53-65.pdf |
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| author | A. Z. Mokhonko A. A. Mokhonko |
| author_facet | A. Z. Mokhonko A. A. Mokhonko |
| author_sort | A. Z. Mokhonko |
| collection | DOAJ |
| description | Let $M_k$ be {the} set of $k$-valued meromorphic in$G={zcolon r_0leqslant |z|}$ functions with {a}~branch point of order$k-1$ {at} $infty$; let $E_ast$ be a set of circles {with finite} sum of radii. Denote$M_ast(r,f)=max|f(z)|, zin{te^{iheta}colon 0leqslanthetaleqslant2kpi,_0leqslant tleqslant r}setminus E_ast, f!in! M_k;$$m(r,f)=frac{1}{2pi k}int_0^{2pik}!ln^+!|f(re^{iheta})|dheta$. If $fin M_k$ is a solution of the equation$P(z,f,f')=0$ and $P$ is a polynomial in all variables theneither $|f(re^{iheta})|<r^u,$ $re^{iheta}in GsetminusE_ast, u>0$ or $m(r,f)$ has growth order$hogeqslantfrac{1}{2k}$, and the following equality holds $lnM_ast(r,f)=(c+o(1))r^ho,$ $ceq0,$ $ro+infty.$ |
| format | Article |
| id | doaj-art-9cdffb3619db407ba5fb64cc433fa151 |
| institution | DOAJ |
| issn | 1027-4634 |
| language | deu |
| publishDate | 2013-10-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-9cdffb3619db407ba5fb64cc433fa1512025-08-20T02:52:09ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342013-10-014015365On growth order of solutions of differential equations in a neighborhood of a branch pointA. Z. MokhonkoA. A. MokhonkoLet $M_k$ be {the} set of $k$-valued meromorphic in$G={zcolon r_0leqslant |z|}$ functions with {a}~branch point of order$k-1$ {at} $infty$; let $E_ast$ be a set of circles {with finite} sum of radii. Denote$M_ast(r,f)=max|f(z)|, zin{te^{iheta}colon 0leqslanthetaleqslant2kpi,_0leqslant tleqslant r}setminus E_ast, f!in! M_k;$$m(r,f)=frac{1}{2pi k}int_0^{2pik}!ln^+!|f(re^{iheta})|dheta$. If $fin M_k$ is a solution of the equation$P(z,f,f')=0$ and $P$ is a polynomial in all variables theneither $|f(re^{iheta})|<r^u,$ $re^{iheta}in GsetminusE_ast, u>0$ or $m(r,f)$ has growth order$hogeqslantfrac{1}{2k}$, and the following equality holds $lnM_ast(r,f)=(c+o(1))r^ho,$ $ceq0,$ $ro+infty.$http://matstud.org.ua/texts/2013/40_1/53-65.pdfalgebraic differential equationbranch point of analytic functionmeromorphic solutionorder of growth |
| spellingShingle | A. Z. Mokhonko A. A. Mokhonko On growth order of solutions of differential equations in a neighborhood of a branch point Математичні Студії algebraic differential equation branch point of analytic function meromorphic solution order of growth |
| title | On growth order of solutions of differential equations in a neighborhood of a branch point |
| title_full | On growth order of solutions of differential equations in a neighborhood of a branch point |
| title_fullStr | On growth order of solutions of differential equations in a neighborhood of a branch point |
| title_full_unstemmed | On growth order of solutions of differential equations in a neighborhood of a branch point |
| title_short | On growth order of solutions of differential equations in a neighborhood of a branch point |
| title_sort | on growth order of solutions of differential equations in a neighborhood of a branch point |
| topic | algebraic differential equation branch point of analytic function meromorphic solution order of growth |
| url | http://matstud.org.ua/texts/2013/40_1/53-65.pdf |
| work_keys_str_mv | AT azmokhonko ongrowthorderofsolutionsofdifferentialequationsinaneighborhoodofabranchpoint AT aamokhonko ongrowthorderofsolutionsofdifferentialequationsinaneighborhoodofabranchpoint |