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Growth estimates for solutions of linear complex differential equations. (English) Zbl 1057.34111
Let \[ f^{(k)}+A_{k-1}(z) f^{(k-1)}+ \cdots+ A_1(z)f'+A_0(z)f=0 \tag{1} \] be a complex differential equation. The coefficients \(A_0(z)\), \(A_1 (z),\dots,A_{k-1}(z)\) are analytic in the disc \(D_R=\{z\in\mathbb{C}: | z| <R\}\), \(0<R \leq+ \infty\). A representation theorem for the solutions of equation (1) is given. By this theorem, for any \(z\), \(z_0 \in D_R\) it holds \[ f(z)=\sum^{K-1}_{n=0} \frac {f^{(n)}(z_0)} {n!}(z-z_0)^n- \frac{1}{(K-1)!} \int^z_{z_0} (z-\xi)^{k-1}A(\xi)f (\xi)\,d\xi. \tag{2} \] The representation thereom yields the growth estimates on the solutions of the general equation (1) in \(D_R\).
(a) If \(0<R\leq 1\), then there exist a constant \(c_1>0\), depending on the initial values of \(f\) at \(z_0=0\), and a constant \(c_2>0\), such that \[ \bigl| f(z)\bigr |\leq c_1\exp\left(c_2\sum^{n-1}_{j=0} \sum^j_{n=0}\int^r_0 \bigl| A_j^{(n)} (se^{i\theta}) \bigr|(R-S)^{K-j+n-1}\,ds \right) \] for all \(\theta\in[0,2\pi)\) and \(r\in[0,R)\).
(b) If \(1<R\leq+ \infty\), then there exist a constant \(C_1> 0\), depending on the initial values of at \(z_0=e^{i\theta}\), and a constant \(C_2>0\), such that \[ \bigl| f(z) \bigr |\leq C_1r^{K-1}\exp \left(C_2 \sum^{K-1}_{j=0} \sum^j_{n= 0} \int^r_0 \bigl| A_j^{(n)}(se^{i\theta}) \bigr | s^{K-j+n-1} \,ds\right) \] for all \(\theta\in [0,a\pi)\) and \(r\in(1,R)\). The Herold’s comparison theorem yields the next growth estimates.

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
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