Some Results on the Deficiencies of Some Differential-Difference Polynomials of Meromorphic Function

For a transcendental meromorphic function f ( z ) , the main aim of this paper is to investigate the properties on the zeros and deficiencies of some differential-difference polynomials. Some results about the deficiencies of some differential-difference polynomials concerning Nevanlinna deficiency and Valiron deficiency are obtained, which are a generalization of and improvement on previous theorems given by Liu, Lan and Zheng, etc.

Fixed Points of Difference Operator of Meromorphic Functions

Letfbe a transcendental meromorphic function of order less than one. The authors prove that the exact differenceΔf=fz+1-fzhas infinitely many fixed points, ifa∈ℂand∞are Borel exceptional values (or Nevanlinna deficiency values) off. These results extend the related results obtained by Chen and Shon.

The distribution of finite values of meromorphic functions with deficient poles

We prove the existence of unbounded open subsets S of the complex plane with the following property. If f is a function transcendental and meromorphic in the plane, the poles of which have positive Nevanlinna deficiency, then f takes every finite value, with at most one exception, infinitely often in the complement of S.

On a conjecture of Fuchs

If f is an entire function of order ρ, 0 < ρ < 2−11, it is shown that the Nevanlinna deficiency d(0, f′/f) of the logarithmic derivative of f satisfies For small positive ρ, this result strengthens an earlier estimate of Eremenko et al. concerning a conjecture of Fuchs.

Oscillation results on y″ + Ay = 0 in the complex domain with transcendental entire coefficients which have extremal deficiencies

Let A(z) be a transcendental entire function and f1, f2 be linearly independent solutions ofWe prove that if A(z) has Nevanlinna deficiency δ(0, A) = 1, then the exponent of convergence of E: = flf2 is infinite. The theorems that we prove here are similar to those in Bank, Laine and Langley [3].

Prime Entire Functions with Prescribed Nevanlinna Deficiency

According to [4] a meromorphic function h(z) = f(g)(z) is said to have f(z) and g(z) as left and right factors respectively, provided that f(z) is non-linear and meromorphic and g(z) is non-linear and entire (g may be meromorphic when f(z) is rational). h(z) is said to be E-prime (E-pseudo prime) if every factorization of the above form into entire factors implies that one of the functions f, or g is linear (polynomial). h(z) is said to be prime (pseudo-prime) if every factorization of the above form, where the factors may be meromorphic, implies that one of f or g is linear (a polynomial or f is rational).