# zbMATH — the first resource for mathematics

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
Full Text:
##### 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 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 New York, reprint of the 1959 edition · Zbl 0322.30001 [7] Yang, L., Value distribution theory, (1993), Springer-Verlag/Science Press Berlin/Beijing, revised edition of the original Chinese edition
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.