We investigate a uniqueness topic of entire functions with finite order sharing some values with their difference operators and obtain one result that if△f(≠0)and entire function f share a,b CM(counting multiplicities),then△f≡f.This result partially confirms a conjecture due to Chen and Yi(2013).
We study the differential equations w2＋ R（z）（w（k））2 = Q（z）, where R（z）, Q（z） are nonzero rationalfunctions. We prove （1） if the differential equation w2 ＋R（z）（w＇）2= Q（z）, where R（z）, Q（z） are nonzero rational functions, admits a transcendental meromorphic solution f, then Q= C （constant）, the multiplicities of the zeros of R（z） are no greater than 2 and f（z） = √C cos α（z）, where α（z） is a primitive of 1 such that √C cos α（z） is a transcendental meromorphic function. （2） if the differential equation w2 ＋ R（z）（w（k））2 = Q（z）, where k ≥ 2 is an integer and R, Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, Q ≡ C （constant）, It（z） ≡ A （constant） and f（z） = √C cos（az ＋ b）, where a2k = A1/A.
Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.
In this paper, we prove the following result： Let f（z） be a transcendental entire function, Q（z） ≡ 0 be a small function of f（z）, and n ≥ 2 be a positive integer. If fn（z） and（fn（z）） share Q（z） CM, then f（z） = ce 1 nz, where c is a nonzero constant. This result extends Lv＇s result from the case of polynomial to small entire function.
