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Entire functions that share values or small functions with their derivatives

Sheng Li;  Zongsheng Gao;  Jilong Zhang — 2012

Annales Polonici Mathematici

We investigate the uniqueness of entire functions sharing values or small functions with their derivatives. One of our results gives a necessary condition on the Nevanlinna deficiency of the entire function f sharing one nonzero finite value CM with its derivative f'. Some applications of this result are provided. Finally, we prove some further results on small function sharing.

Distribution of zeros and shared values of difference operators

Jilong Zhang;  Zongsheng Gao;  Sheng Li — 2011

Annales Polonici Mathematici

We investigate the distribution of zeros and shared values of the difference operator on meromorphic functions. In particular, we show that if f is a transcendental meromorphic function of finite order with a small number of poles, c is a non-zero complex constant such that Δ c k f ≠ 0 for n ≥ 2, and a is a small function with respect to f, then f ⁿ Δ c k f equals a (≠ 0,∞) at infinitely many points. Uniqueness of difference polynomials with the same 1-points or fixed points is also proved.

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