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.
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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 $Δ^k_cf ≠ 0$ for n ≥ 2, and a is a small function with respect to f, then $fⁿΔ^k_cf$ 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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