Growth and oscillation of some class of differential polynomials in the unit disc

Benharrat Belaïdi

Kragujevac Journal of Mathematics, Tome 35 (2011) no. 3, p. 369

Résumé

In this paper, we study the growth and the oscillation of complex differential equations $f^{^{\prime \prime}}+A_{1}\left( z\right) f^{^{\prime }}+A_{0}\left( z\right) f=0$ and $%f^{^{\prime \prime }}+A_{1}\left( z\right) f^{^{\prime }}+A_{0}\left( z\right) f=F,$ where $A_{0}\not\equiv 0$, $A_{1}$ and $F$ are analytic functions in the unit disc $\Delta =\left\{z:\left\vert z\right\vert 1\right\} $ with finite iterated $p-$order. We obtain some results on the iterated $p-$order and the iterated exponent of convergence of zero-points in $\Delta $ of the differential polynomials $%g_{f}=d_{1}f^{^{\prime }}+d_{0}f$ and $g_{f}=d_{1}f^{^{\prime }}+d_{0}f+b$, where $d_{1},d_{0},b$ are analytic functions such that at least one of $% d_{0}\left( z\right) ,d_{1}\left( z\right) $ does not vanish identically with $\rho _{p}\left( d_{j}\right) \infty$ $\left(j=0,1\right) ,\rho _{p}\left( b\right) \infty$.

Détail

Classification : 34M10 30D35
Keywords: Linear differential equations, analytic function, iterated $p-$order, iterated exponent of convergence of the sequence of distinct zeros.