Meromorphic function sharing a small function with a linear differential polynomial

Lahiri, Indrajit ; Sarkar, Amit

Mathematica Bohemica, Tome 141 (2016) no. 1, pp. 1-11.

Résumé

The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and prove the following result: Let $f$ be a nonconstant meromorphic function and $L$ a nonconstant linear differential polynomial generated by $f$. Suppose that $a = a(z)$ ($\not \equiv 0, \infty $) is a small function of $f$. If $f-a$ and $L-a$ share $0$ CM and \[ (k+1)\overline N(r, \infty ; f)+ \overline N(r, 0; f')+ N_{k}(r, 0; f') \lambda T(r, f')+ S(r, f') \] for some real constant $\lambda \in (0, 1)$, then $ f-a=(1+ {c}/{a})(L-a)$, where $c$ is a constant and $1+{c}/{a} \not \equiv 0$.

Détail

DOI : 10.21136/MB.2016.1

Classification : 30D35
Keywords: meromorphic function; differential polynomial; small function; sharing