an:01997636
Zbl 1044.34049
Chyzhykov, Igor; Gundersen, Gary G.; Heittokangas, Janne
Linear differential equations and logarithmic derivative estimates
EN
Proc. Lond. Math. Soc., III. Ser. 86, No. 3, 735-754 (2003).
00097878
2003
j
34M10 30D35 34M05 34M45
linear differential equation; logarithhmic derivatives estimates; meromorphic functions; analytic functions; method of successive approximations; Nevanlinna theory
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 [\textit{H. Wittich}, Neuere Untersuchungen ??ber 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.
Natalia Medvedeva (Chelyabinsk)
Zbl 0159.10103