Abstract
This paper is to consider the unity results on entire functions sharing two values with their difference operators and to prove some results related to 4 CM theorem. The main result reads as follows: Let f(z) be a nonconstant entire function of finite order, and let a_{1}, a_{2} be two distinct finite complex constants. If f(z) and Delta _{eta }^{n}f(z) share a_{1} and a_{2} “CM”, then f(z)equiv Delta _{eta }^{n} f(z), and hence f(z) and Delta _{eta }^{n}f(z) share a_{1} and a_{2} CM.
Highlights
1 Introduction and main results It is well known that a monic polynomial is uniquely determined by its zeros and a rational function by its zeros and poles ignoring a constant factor
It becomes much more complicated to deal with the transcendental meromorphic function case
And in what follows, we say that f (z) and g(z) share the finite value a CM(IM) if f (z) – a and g(z) – a have the same zeros with the same multiplicities, and we say that f (z) and g(z) share the ∞ CM(IM) if f (z) and g(z) have the same poles with the same multiplicities
Summary
– NE(r, a) = S(r, g), where NE(r, a) is defined to be the reduced counting function of common zeros of f (z) – a and g(z) – a with the same multiplicities. NE1)(r, a) used later is defined to be the reduced counting function of common simple zeros of f (z) – a and g(z) – a. Applying Theorem A, one can get (see Theorem 4.8 in [23]) the following. Theorem B ([23]) Let f and g be nonconstant meromorphic functions and aj (j = 1, 2, 3, 4)
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