On Petrenko's deviations and second order differential equations

Abstract

New results on the oscillation of solutions of $f^{\prime \prime }+A(z)f=0$ and on the growth of solutions of $f^{\prime \prime }+A(z)f^{\prime }+B(z)f=0$ are obtained, where $A$ and $B$ are entire functions. Petrenko's magnitudes of deviation of $g$ with respect to $\infty$ play a key rôle in the results, where $g$ represents one of the coefficients $A$ or $B$. These quantities are defined by ${\beta }^{-}(\infty ,g)={\mathrm{liminf}}_{r\to \infty }\frac{\log M(r,g)}{T(r,g)}$ and ${\beta }^{+}(\infty ,g)={\mathrm{limsup}}_{r\to \infty }\frac{\log M(r,g)}{T(r,g)}$.

9 pages