# zbMATH — the first resource for mathematics

On the complex oscillation theory of $$f^{\prime \prime } + A (z)f = 0$$ where $$A (z)$$ is analytic in the unit disc. (English) Zbl 1193.34174
The authors study the complex oscillation theory of the second linear differential equation
$f''+ A(z) f= 0,\tag{1}$ where $$A(z)$$ is analytic in the unit disk $$D= \{z:|z|< 1\}$$. The order of meromorphic function $$f$$ in $$D$$ can be defined either as
$\sigma(f):= \limsup_{r\to 1-} {\log^+T(r, f)\over-\log(1- r)},$ where $$T(r,f)$$ is the Nevanlinna characteristic of $$f$$. Let $$\lambda(f)$$ denote the exponents of convergence of the sequence of zeros of the function $$f$$, $$\overline\lambda(f)$$ to denote the respectively the exponents of convergence of the sequence of distinct zeros of $$f$$, and use the notation $$\sigma_2(f)$$ to denote the hyper-order of $$f(z)$$. For analytic function $$f$$ in $$D$$, we also define
$\sigma_M(f):= \limsup_{r\to 1-} {\log^+\log^+ M(r,f)\over -\log(1- r)}.$ The authors prove the following theorems:
Theorem 1. Let $$A(z)$$ be an admissible analytic function in the unit disc $$D$$. Then all nonzero solutions $$f$$ of equation (1) are of infinite order and satisfy $$\sigma(A)\leq\sigma_2(f)=\sigma_M(A)$$.
Theorem 2. Let $$A(z)$$ be an admissible analytic function in the unit disc $$D$$. If $$\overline\lambda(A)<\sigma(A)$$, then every nonzero solution $$f$$ of equation (1) satisfies $$\sigma(A)\leq\overline\lambda(f)$$.

##### 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
Full Text:
##### References:
 [1] Bank, A general theorem concerning the growth of solutions of first-order algebraic differential equations, Compos. Math. 25 pp 61– (1972) · Zbl 0246.34006 [2] Bank, On the oscillation theory of f ” + Af = 0 where A is entire, Trans. Amer. Math. Soc. 273 pp 351– (1982) · Zbl 0505.34026 [3] Bank, On the zeros of meromorphic solutions of second order linear differential equations, Comment. Math. Helv. 58 pp 656– (1983) [4] Chen, The growth of solutions of differential equations with coefficients of small growth in the disc, J. Math. Anal. Appl. 297 pp 285– (2004) · Zbl 1062.34097 [5] Chyzhykov, Linear differential equations and logrithmic derivative estimates, Proc. London Math. Soc. (3) 86 pp 735– (2003) [6] W. Hayman, Meromorphic Functions (Clarendon Press, Oxford, 1964). · Zbl 0115.06203 [7] Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss. 122 pp 1– (2000) · Zbl 0965.34075 [8] Heittokangas, Growth estimates for solutions of linear complex differential equations, Ann. Acad. Sci. Fenn. Math. 29 pp 233– (2004) · Zbl 1057.34111 [9] I. Laine, Nevanlinna Theory and Complex Differential Equations (W. de Gruyter, Berlin, 1993). [10] Li, On the growth of the solution of two-order differential equations in the unit disc, Pure Appl. Math. (Xi’an) 4 pp 295– (2002) [11] Pommenrenke, On the mean growth of solutions of complex linear differential equations in the disk, Complex Variables Elliptic Equations 1 (1) pp 23– (1982) [12] M. Tsuji, Potential Theory in Modern Function Theory (Chelsea, New York, 1975) (reprint of the 1959 edition). · Zbl 0322.30001 [13] L. Yang, Value Distribution Theory (Springer-Verlag, Berlin, 1993; Science Press, Beijing, 1993). · Zbl 0790.30018 [14] C.-C. Yang, and H.-X. Yi, Uniqueness Theory of Meromorphic Functions (Science Press, Beijing, 1995/Kluwer, Dordrecht, 2003).
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.