Asymptotic properties of meromorphic solutions of differential equations in a neighborhood of a logarithmic singularity

L. I. Kolyasa, A. Z. Mokhon’ko, V. D. Mokhon’ko
Lviv Polytechnic National University
We obtain asymptotic estimates of the moduli of meromorphic solutions with a logarithmic singularity at $\infty$ of the differential equation \begin{gather*} \sum\limits_{k+s=m}f^kf_1^sv_{ks}(z)z^{\tau_{ks}}\text{Ln} ^{\varkappa_{ks}}z=\sum\limits_{|K|\leq m}b_K(z)f^{k_0}f_1^{k_1}\dots f_p^{k_p}, \\ f'=f_1, \ldots,f^{(p)}=f_p,\ K=(k_0,k_1,\ldots,k_p),\ |K|=k_0+k_1+\ldots+k_p;\\ \tau_{m-s,s}-s\leqslant\tau_{m-n,n}-n,\ s\leq n=\max\{s\colon k+s=m, c_{ks}\neq0\}; \end{gather*} where $v_{ks}(z), b_K(z)$ are analytic functions such that $\forall \alpha, \beta \in\mathbb{R},$ \[ \begin{array}{c} |b_K(re^{i\theta})|\leq r^{\tau_K},\ v_{ks}(re^{i\theta})=c_{ks}+o(1),\ r\to+\infty,\ \alpha\leqslant\theta\leqslant\beta;\ \tau_{ks},\ \varkappa_{ks},\ \tau_K\in\mathbb{R},\ c_{ks}\in\mathbb{C}. \end{array} \]
meromorphic function; logarithmic singularity; differential equation
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