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.