Estimates of derivatives of meromorphic functions on sets of a-points. (English) Zbl 0608.30034

In the paper,the following result is proved: Suppose that w is meromorphic in \({\mathbb{C}}\), that \(a_{\nu}\), \(1\leq \nu \leq q\), \(q>4\), are different complex constants, and that \(\psi(r)\) is monotone decreasing on \([0,\infty)\) with \(\psi(r)\to 0\) as \(r\to \infty\). Then there is a set E of finite logarithmic measure in \([0,\infty)\) such that for every \(r\not\in E\) there is a subset \(\{z_{\nu,k}\}\), \(1\leq k\leq n_ 0(r,a_{\nu})\) of the \(a_{\nu}\)-points of w which lie in \(| z| \leq r\) and, moreover, satisfy \[ | w'(z_{\nu,k})| \geq \psi (r)A^{1/2}(r)/r,\quad 1\leq \nu \leq q,\quad 1\leq k\leq n_ 0(r,a_{\nu}), \]
\[ \sum^{q}_{\nu =1}n_ 0(r,a_{\nu})\geq (q- 4)A(r)-o[A(r)],\quad r\to \infty,\quad r\not\in E. \] This result gives easy estimates of the order of meromorphic solutions of some classes of algebraic differential equations, which is elegantly demonstrated in the closing part of the paper.
Reviewer: A.Klić


30D30 Meromorphic functions of one complex variable (general theory)
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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