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

  • W. Lü China University of Petroleum
  • B. CHAKRABORTY Ramakrishna Mission Vivekananda Centenary College
Keywords: value distribution theory;, transcendental meromorphic function;, differential monomials;, normal family

Abstract

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 $D$.

Author Biographies

W. Lü, China University of Petroleum

China University of Petroleum

B. CHAKRABORTY, Ramakrishna Mission Vivekananda Centenary College

Ramakrishna Mission Vivekananda Centenary College

References

H.Y. Chen, M.L. Fang, The value distribution of ff′, Sci. China Math., 38 (1995), 789–798.

W.K. Hayman, Meromorphic functions, The Clarendon Press, Oxford, 1964.

W.K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. Math., 70 (1959), 9–42.

X. Huang, Y. Gu, On the value distribution of $f^{2}f^{(k)}$, J. Aust. Math. Soc., 78 (2005), 17–26.

Y. Jiang, B. Huang, A note on the value distribution of $f^{l}(f^{(k)})^{n}$, Arxiv: 1405.3742v1 [math.CV] 15 May 2014.

I. Lahiri, S. Dewan, Inequalities arising out of the value distribution of a differential monomial, J. Inequal. Pure Appl. Math., 4 (2003), №2, Article 27.

N. Li, L.Z. Yang, Meromorphic function that shares one small functions with its differential polynomial, Kyungpook Math. J., 50 (2010), 447–454.

E. Mues, ¨ Uber ein Problem von Hayman, Math. Z., 164 (1979), 239–259.

X.C. Pang, Bloch’s principle and normal criterion, Sci. China Ser. A, 33 (1989), 782–791.

J. Schiff, Normal families, Springer, New York, 1993.

N. Steinmetz, Nevanlinna theory, normal families, and algebraic differential equations, Springer, 2017.

J.F. Xu, H.X. Yi, Z.L. Zhang, Some inequalities of differential polynomials, Math. Inequal. Appl., 12

(2009), 99–113.

J.F. Xu, H.X. Yi, Z.L. Zhang, Some inequalities of differential polynomials II, Math. Inequal. Appl., 14 (2011), 93–100.

J.F. Xu, H.X. Yi, A precise inequality of differential polynomials related to small functions, Journal of mathematical inequalities, 10 (4) (2016), 971–976.

J. Xu, S. Ye, On the zeros of the differential polynomial $varphi(z) f(z)^{2} (f'(z))^{2}- 1$, Mathematics, doi:10.3390/math7010087.

Q.D. Zhang, A growth theorem for meromorphic functions, J. Chengdu Inst. Meteor., 20 (1992), 12–20.

Published
2021-10-23
How to Cite
LüW., & CHAKRABORTY, B. (2021). On the value distribution of a differential monomial and some normality criteria. Matematychni Studii, 56(1), 55-60. https://doi.org/10.30970/ms.56.1.55-60
Section
Articles