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Linear differential equations and logarithmic derivative estimates. (English) Zbl 1044.34049
The linear differential equation \[ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\ldots+A_0(z)f=0 \tag{1} \] is considered, where \(A_n(z)\), \(n=0,1,\dots,k-1\), are analytic functions in the unit disk \(\Delta= \{z: | z|<1 \}\) in the complex plane. Two sharp inequalities for the growth of solutions of certain equations of the form (1) are proved. The obtained results are analogous to the results from [H. Wittich, Neuere Untersuchungen über eindeutige analytische Funktionen, Ergebnisse der Mathematik und ihrer Grenzgebiete, 8, 2. korrig. Aufl. Berlin: Springer (1968; Zbl 0159.10103), Chapter 5, 3] concerning the solutions of equation (1) with polynomial coefficients in the whole complex plane.
To prove the mentioned inequalities, the method of successive approximations and sharp estimates on the logarithmic derivatives of finite-order meromorphic functions in the unit disc are used.

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
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
34M45 Ordinary differential equations on complex manifolds
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