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On subnormal solutions of second order linear periodic differential equations. (English) Zbl 1132.34064

The authors study the growth of solutions of the linear periodic differential equation

${f}^{\text{'}\text{'}}+\left({P}_{1}\left({e}^{z}\right)+{P}_{2}\left({e}^{-z}\right)\right){f}^{\text{'}}+\left({Q}_{1}\left({e}^{z}\right)+{Q}_{2}\left({e}^{-z}\right)\right)f={R}_{1}\left({e}^{z}\right)+{R}_{2}\left({e}^{-z}\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

and its corresponding homogeneous equation

${f}^{\text{'}\text{'}}+\left({P}_{1}\left({e}^{z}\right)+{P}_{2}\left({e}^{-z}\right)\right){f}^{\text{'}}+\left({Q}_{1}\left({e}^{z}\right)+{Q}_{2}\left({e}^{-z}\right)\right)f=0,\phantom{\rule{2.em}{0ex}}\left(2\right)$

where ${P}_{j}\left(z\right),{Q}_{j}\left(z\right),{R}_{j}\left(z\right)$ $\left(j=1,2\right)$ are polynomials in $z$. It is well-known that all solution of (2) are entire functions. Let $\sigma \left(f\right)$ denote the order of growth of $f\left(z\right)$ and use the notation ${\sigma }_{2}\left(f\right)$ to denote the hyper-order of $f\left(z\right)$.

Theorem 1. If

$deg{Q}_{1}>deg{P}_{1}\phantom{\rule{1.em}{0ex}}\text{or}\phantom{\rule{1.em}{0ex}}deg{Q}_{2}>deg{P}_{2},\phantom{\rule{2.em}{0ex}}\left(3\right)$

then the differential equation (2) has no nontrivial subnormal solution, and every solution of (2) satisfies ${\sigma }_{2}\left(f\right)=1$.

Theorem 2. If ${R}_{1}+{R}_{2}\ne 0$ and ${P}_{j},{Q}_{j}$ ($j=1,2$) satisfy (3), then (i) (1) has at most one subnormal solution ${f}_{0}$, and ${f}_{0}={S}_{1}\left({e}^{z}\right)+{S}_{2}\left({e}^{-z}\right)$, where ${S}_{1}\left(z\right)$ and ${S}_{2}\left(z\right)$ are polynomials in $z$; (ii) all other solutions $f$ of (1) satisfy ${\sigma }_{2}\left(f\right)=1$ except a subnormal solution in (i).

Theorem 2 refines and generalizes a result due to G. Gundersen and M. Steinbart for ${P}_{2}={Q}_{2}=0$.

##### MSC:
 34M10 Oscillation, growth of solutions (ODE in the complex domain)
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