Abstract
In this paper, we study the uniqueness problem of entire functions sharing polynomials IM with their first derivative. As an application, we generalize Brück’s conjecture from sharing value CM to sharing polynomial IM for a class of functions. In fact, we prove a result as follows: Let \({a({\not\equiv} 0)}\) be a polynomial and \({n \geq 2}\) be an integer, let f be a transcendental entire function, and let \({F = f^n}\). If F and F′ share a IM, then \({f(z) = Ae^{z/n},}\) where A is a nonzero constant. It extends some previous related theorems.
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The paper was supported by the Natural Science Foundation of Shandong Province Youth Fund Project (ZR2012AQ021) and National Natural Science Foundation of China (11171184).
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Lü, F., Yi, H. A Generalization of Brück’s Conjecture for a Class of Entire Functions. Results. Math. 68, 157–169 (2015). https://doi.org/10.1007/s00025-014-0428-9
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DOI: https://doi.org/10.1007/s00025-014-0428-9