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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 \[ {a_{k,d}\over k-d}- 2\leq \sigma(f)\leq \sigma_M(f)\leq \max\{a_{k,j}- 1; j= 1,\dots,k-1\}. \]

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
Full Text: DOI
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