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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| Main Authors: | , |
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| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2013-10-01
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| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/texts/2013/40_1/53-65.pdf |
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| Summary: | 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 $lnM_ast(r,f)=(c+o(1))r^ho,$ $ceq0,$ $ro+infty.$ |
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| ISSN: | 1027-4634 |