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Growth estimates for solutions of linear complex differential equations. (English) Zbl 1057.34111

Let

${f}^{\left(k\right)}+{A}_{k-1}\left(z\right){f}^{\left(k-1\right)}+\cdots +{A}_{1}\left(z\right){f}^{\text{'}}+{A}_{0}\left(z\right)f=0\phantom{\rule{2.em}{0ex}}\left(1\right)$

be a complex differential equation. The coefficients ${A}_{0}\left(z\right)$, ${A}_{1}\left(z\right),\cdots ,{A}_{k-1}\left(z\right)$ are analytic in the disc ${D}_{R}=\left\{z\in ℂ:|z|, $0. A representation theorem for the solutions of equation (1) is given. By this theorem, for any $z$, ${z}_{0}\in {D}_{R}$ it holds

$f\left(z\right)=\sum _{n=0}^{K-1}\frac{{f}^{\left(n\right)}\left({z}_{0}\right)}{n!}{\left(z-{z}_{0}\right)}^{n}-\frac{1}{\left(K-1\right)!}{\int }_{{z}_{0}}^{z}{\left(z-\xi \right)}^{k-1}A\left(\xi \right)f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}d\xi ·\phantom{\rule{2.em}{0ex}}\left(2\right)$

The representation thereom yields the growth estimates on the solutions of the general equation (1) in ${D}_{R}$.

(a) If $0, 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

$\left|f\left(z\right)\right|\le {c}_{1}exp\left({c}_{2}\sum _{j=0}^{n-1}\sum _{n=0}^{j}{\int }_{0}^{r}\left|{A}_{j}^{\left(n\right)}\left(s{e}^{i\theta }\right)\right|{\left(R-S\right)}^{K-j+n-1}\phantom{\rule{0.166667em}{0ex}}ds\right)$

for all $\theta \in \left[0,2\pi \right)$ and $r\in \left[0,R\right)$.

(b) If $1, 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

$\left|f\left(z\right)\right|\le {C}_{1}{r}^{K-1}exp\left({C}_{2}\sum _{j=0}^{K-1}\sum _{n=0}^{j}{\int }_{0}^{r}\left|{A}_{j}^{\left(n\right)}\left(s{e}^{i\theta }\right)\right|{s}^{K-j+n-1}\phantom{\rule{0.166667em}{0ex}}ds\right)$

for all $\theta \in \left[0,a\pi \right)$ and $r\in \left(1,R\right)$. The Herold’s comparison theorem yields the next growth estimates.

##### MSC:
 34M10 Oscillation, growth of solutions (ODE in the complex domain) 30D35 Distribution of values (one complex variable); Nevanlinna theory