On the value distribution of a differential monomial and some normality criteria

W. Lü; B. CHAKRABORTY

doi:10.30970/ms.56.1.55-60

W. Lü China University of Petroleum
B. CHAKRABORTY Ramakrishna Mission Vivekananda Centenary College

DOI: https://doi.org/10.30970/ms.56.1.55-60

Анотація

The aim of this paper is to study the zero distribution of the differential polynomial

$\displaystyle af^{q_{0}}(f')^{q_{1}}...(f^{(k)})^{q_{k}}-\varphi,$

where $f$ is a transcendental meromorphic function and $a=a(z)(\not\equiv 0,\infty)$ and $\varphi(\not\equiv 0,\infty)$ are small functions of $f$. Moreover, using this value distribution result, we prove the following normality criterion for family of analytic functions:\\ {\it Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k \geq1$, $q_{0}\geq 2$, $q_{i} \geq 0$ $(i=1,2,\ldots,k-1)$, $q_{k}\geq 1$ be positive integers. If for each $f\in \mathscr{F}$: i.\ $f$ has only zeros of multiplicity at least $k$,
\ ii.\

$\displaystyle f^{q_{0}}(f')^{q_{1}}\ldots(f^{(k)})^{q_{k}}\not=1$,

then $\mathscr{F}$ is normal on domain $D$.

Біографії авторів

W. Lü, China University of Petroleum

China University of Petroleum

B. CHAKRABORTY, Ramakrishna Mission Vivekananda Centenary College

Ramakrishna Mission Vivekananda Centenary College

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