Abstract
AbstractA result of Hinchliffe (2003) is extended to transcendental entire function, and an alternative proof is given in this paper. Our main result is as follows: let "Equation missing" be an analytic function, "Equation missing" a family of analytic functions in a domain "Equation missing", and "Equation missing" a transcendental entire function. If "Equation missing" and "Equation missing" share "Equation missing" IM for each pair "Equation missing", and one of the following conditions holds: (1) "Equation missing" has at least two distinct zeros for any "Equation missing"; (2) "Equation missing" is nonconstant, and there exists "Equation missing" such that "Equation missing" has only one distinct zero "Equation missing", and suppose that the multiplicities "Equation missing" and "Equation missing" of zeros of "Equation missing" and "Equation missing" at "Equation missing", respectively, satisfy "Equation missing", for each "Equation missing", where "Equation missing"; (3) there exists a "Equation missing" such that "Equation missing" has no zero, and "Equation missing" is nonconstant, then "Equation missing" is normal in "Equation missing".
Highlights
Introduction and Main ResultsLet f z and g z be two nonconstant meromorphic functions in the whole complex plane C, and let a be a finite complex value or function
It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna’s value-distribution theory
We denote by S r, f any function satisfying S r, f possibly outside of a set of finite measure
Summary
Let f z and g z be two nonconstant meromorphic functions in the whole complex plane C, and let a be a finite complex value or function. Let P z be a polynomial of degree p at least 2 and f z a transcendental entire function, and α z a nonconstant meromorphic function satisfying T r, α S r, f. Yuan et al 10 generalized Theorem G in another manner and proved the following result. Let α z be a nonconstant meromorphic function, F a family of analytic functions in a domain D, and R z a rational function of degree at least 2. If H ◦ f z and H ◦ g z share α z IM for each pair f z , g z ∈ F, and one of the following conditions holds:
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