On global solutions of complex differential equations in the unit disk. (English) Zbl 1213.30053

Benbourenane, D.; Sons, L. R.

Complex Variables, Theory Appl. 49, No. 13, 913-925 (2004).

By using Nevanlinna value distribution theory, many authors considered the properties of solutions of linear differential equations with meromorphic coefficients in complex domains. However, there are less results about the properties of solutions of linear differential equations with meromorphic coefficients in the unit disk. In this paper, the authors consider the homogeneous linear differential equation \[ f^{(n)}+a_{n-1}(z)f^{(n-1)}+\cdots+a_1 (z)f^{'}+a_0 (z)f=0, \qquad\qquad(*) \] where the coefficients \(a_j (z)\) (\(j=0,1,\cdots,n-1\)) are analytic in the unit disk. They obtain the properties of any solution of (*), which extends the results obtained by G. Valiron and S. Bank in [G. Valiron, J. Math. Pures Appl., IX. Sér. 31, 293–303 (1952; Zbl 0047.31003)] and [S. Bank, Ann. Mat. Pura Appl., IV. Ser. 92, 323–335 (1972; Zbl 0246.34004)] for the case when the coefficients are polynomials. Moreover, some interaction between the growth of the coefficients and the growth of solutions of equation (*) are presented.

Reviewer: Zhibo Huang (Guangzhou)

MSC:

30D35

Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Keywords:

Small growth; Fundamental system; Wronskian; Finite order; Nevanlinna characteristic