Complex oscillation of differential polynomials in the unit disc.

*(English)*Zbl 1299.34284Summary: 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 |

##### Keywords:

linear differential equations; analytic function; hyper-order; exponent of convergence of the sequence of distinct zeros
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\textit{Z. Latreuch} et al., Period. Math. Hung. 66, No. 1, 45--60 (2013; Zbl 1299.34284)

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