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Complex oscillation of differential polynomials in the unit disc. (English) Zbl 1299.34284
Summary: We consider the complex differential equations \[ f''+A_1(z)f'+A_0(z)f = F, \] where \(A_0\not\equiv 0\), \(A_1\) and \(F\) are analytic functions in the unit disc \(\Delta =\{z : | z| < 1\}\). We obtain results on the order and the exponent of convergence of zero-points in \(\Delta\) of the differential polynomials \(g_f = d_2 f'' + d_1 f' +d_ 0f\) with non-simultaneously vanishing analytic coefficients \(d_2\), \(d_1\), \(d_0\). We answer a question posed by J. Tu and C. F. Yi [J. Math. Anal. Appl. 340, No. 1, 487–497 (2008; Zbl 1141.34054)] for the case of second-order linear differential equations in the unit disc.

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
34M03 Linear ordinary differential equations and systems in the complex domain
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