On the value distribution of a Differential Monomial and some normality criteria

Abstract

Let $f$ be a transcendental meromorphic function defined in the complex plane $\mathbb{C}$, and $\phi (\not\equiv 0,\infty )$ be a small function of $f$. In this paper, We give a quantitative estimation of the characteristic function $T(r,f)$ in terms of $N\left(r,\frac{1}{M[f]-\phi (z)}\right)$ as well as $\ol{N}\left(r,\frac{1}{M[f]-φ(z)}\right)$, where $M[f]$ is the differential monomial, generated by $f$.\par Moreover, we prove one normality criterion: Let $\mathcal{F}$ be a family of analytic functions on a domain $D$ and let $k(\ge 1)$, $q_0(\ge 3)$, $q_i(\ge 0)$ $(i=1,2,\dots ,k-1)$, $q_k(\ge 1)$ be positive integers. If for each $f\in \mathcal{F}$, $f$ has only zeros of multiplicity at least $k$, and $f^{q_0}(f^{\prime }{)}^{q_1}...(f^{(k)}{)}^{q_k}\ne 1$, then $\mathcal{F}$ is normal on domain $D$.

8 pages