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The growth of solutions of differential equations with coefficients of small growth in the disc. (English) Zbl 1062.34097
Let $$f$$ be analytic in $$\Delta= \{z:| z|< 1\}$$. If $$\varlimsup_{r\to 1^-} {\log M(r\cdot f)\over\log{1\over 1-\gamma}}= b<\infty$$ (or $$=\infty$$), then say $$f$$ is a function of finite $$b$$ degree, and denote it by $$f\in D_b$$. If $$\varlimsup_{r\to 1^-}{\log M(r\cdot f)\over\log{1\over 1-\gamma}}\leq b< \infty$$, one say that, $$f$$ is a function of small growth $$f\in SD_b$$. Let $f^{(k)}+ A_{k-1} f^{(k-1)}+\cdots+ A_0 f= 0\tag{*}$ be a linear differential equation. The authors obtain the following main result:
Let $$A_j$$, $$j= 0,\dots, k-1$$, be analytic in $$\Delta$$. Suppose that there exists some $$d\in\{0,\dots, k-1\}$$ such that $$A_d\in D_{a_{k,d}}$$ ($$a_{k,d}$$ is a positive real number), while $$A_j\in SD_{a_{k,j}}$$ ($$j\neq d$$, $$a_{k,j}$$ are nonnegative real numbers), and such that ${a_{k,d}\over k-d}> \max\Biggl\{2,\,{a_{k,j}\over k-j}\;(j\neq d)\Biggr\}.$ Then: (i) (*) possesses at most $$d$$ linearly independent analytic solutions of zero-order. Thus, at most $$d$$ linearly independent finite degree analytic solutions. (ii) If (*) has $$d$$ linearly independent finite degree analytic solutions, then (*) must have $$k-d$$ linearly independent analytic solutions satisfying ${a_{k,d}\over k-d}- 2\leq \sigma(f)\leq \sigma_M(f)\leq \max\{a_{k,j}- 1; j= 1,\dots,k-1\}.$

##### MSC:
 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
growth order
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##### References:
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