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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

Keywords:

growth order
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References:

[1] Chyzhykov, I.; Gundersen, G.; Heittokangas, J., Linear differential equations and logarithmic derivative estimates, Proc. London Math. Soc., 86, 735-754 (2003) · Zbl 1044.34049
[2] Frei, M., Über die lösungen linearer differentialgleichungen mit ganzen funktionen als koeffizienten, Comment. Math. Helv., 35, 201-222 (1961) · Zbl 0115.06903
[3] Hayman, W., Meromorphic Functions (1964), Clarendon: Clarendon Oxford · Zbl 0115.06203
[4] Heittokangas, J., On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss., 122, 1-54 (2000) · Zbl 0965.34075
[5] Pommerenke, Chr., On the mean growth of the solutions of complex linear differential equations in the disk, Complex Variables, 1, 23-38 (1982) · Zbl 0464.34010
[6] Tsuji, M., Potential Theory in Modern Function Theory (1975), Chelsea: Chelsea New York, reprint of the 1959 edition · Zbl 0322.30001
[7] Yang, L., Value Distribution Theory (1993), Springer-Verlag/Science Press: Springer-Verlag/Science Press Berlin/Beijing, revised edition of the original Chinese edition
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