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The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc. (English) Zbl 1105.34059

The authors study the meromorphic solutions of a linear differential equation in the unit disc. They first give a definition of iterated order of a meromorphic function in the unit disc. They define the iterated \(n\)th-order \(\sigma_n(f)\) of the meromorphic function \(f\) as \[ \sigma_n(f)=\limsup_{r\to\infty}\frac{\log^{[n]}T(r,f)}{\log{\frac1{1-r}}},\quad n\in{\mathbb{N}}, \] and the growth index of the iterated order of a meromorphic function \(f(z)\) in the unit disc as \[ i(f)= \begin{cases} 0 &\text{if \(f\) is nonadmissible}; \\ \min\{n: \sigma_n(f)<\infty\} &\text{if \(f\) is admissible};\\ \infty &\text{if } \sigma_n(f)=\infty \text{ for all } n\in \mathbb{N}. \\ \end{cases} \] Then, they investigate the iterated order of analytic solutions of the following linear differential equation with analytic coefficients in the unit disc \[ f^{(k)}+a_{k-1}(z)f^{(k-1)}+\cdots +a_0(z)f=0. \] For example, they prove that if \(f\) is an analytic solution of this equation, then \(i(f)\leq \max\{i(a_j)\), \(j=1,\cdots, k-1\}.\)

MSC:

34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D30 Meromorphic functions of one complex variable (general theory)
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[1] Bernal, L. G., On growth \(k\)-order of solutions of a complex homogeneous linear differential equations, Proc. Amer. Math. Soc., 101, 317-322 (1987) · Zbl 0652.34008
[2] Chyzhykov, I.; Gundersen, G.; Heittokangas, J., Linear differential equations and logarithmic derivative estimates, Proc. London Math. Soc., 86, 735-754 (2003) · Zbl 1044.34049
[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] Tsuji, M., Potential Theory in Modern Function Theory (1975), Chelsea: Chelsea New York, reprint of the 1959 edition · Zbl 0322.30001
[6] Yang, L., Value Distribution Theory (1993), Springer-Verlag/Science Press: Springer-Verlag/Science Press Berlin/Beijing
[7] Chen, Z.-X.; Shon, K. H., The growth of solutions of differential equations with coefficients of small growth in the disc, J. Math. Anal Appl., 297, 285-304 (2004) · Zbl 1062.34097
[8] Li, Y.-Z., On the growth of the solution of two-order differential equations in the unit disc, Pure Appl. Math., 4, 295-300 (2002) · Zbl 1128.34330
[9] Chen, Z.-X., The properties of solutions of one certain differential equation in the unit disc, J. Jiangxi Norm. Univ., 3, 189-190 (2002)
[10] Wittich, H., Zur Theorie linearer Differentialgleichungen in Komplexen, Ann. Acad. Sci. Fenn. Ser. A I, 379, 1-18 (1966) · Zbl 0139.03602
[11] Kinnunen, L., Linear differential equations with solutions of finite iterated order, Southeast Asian Bull. Math., 22, 4, 385-405 (1998) · Zbl 0934.34076
[12] Chuang, C. T., Sur la comparaison de la croissance d’une fonction méromorphe de celle de sa dérivée, Bull. Sci. Math., 75, 171-190 (1951) · Zbl 0045.35703
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