LOWER ESTIMATES FOR THE GROWTH OF THE FOURTH AND THE SECOND PAINLEVÉ TRANSCENDENTS
LOWER ESTIMATES FOR THE GROWTH OF THE FOURTH AND THE SECOND PAINLEVÉ TRANSCENDENTS
Abstract Let $w(z)$ be an arbitrary transcendental solution of the fourth (respectively, second) Painlevé equation. Concerning the frequency of poles in $|z|\le r$, it is shown that $n(r,w)\gg r^2$ (respectively, $n(r,w)\gg r^{3/2}$), from which the growth estimate $T(r,w)\gg r^2$ (respectively, $T(r,w)\gg r^{3/2}$) immediately follows. AMS 2000 Mathematics subject classification: Primary …