Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives

Jianming Qi, Feng Lü and Ang Chen

Abstract and Applied Analysis, 2009, vol. 2009, issue 1

Abstract: We use the theory of normal families to prove the following. Let Q1(z) = a1zp + a1,p−1zp−1 + ⋯+a1,0 and Q2(z) = a2zp + a2,p−1zp−1 + ⋯+a2,0 be two polynomials such that deg Q1 = deg Q2 = p (where p is a nonnegative integer) and a1, a2(a2 ≠ 0) are two distinct complex numbers. Let f(z) be a transcendental entire function. If f(z) and f′(z) share the polynomial Q1(z) CM and if f(z) = Q2(z) whenever f′(z) = Q2(z), then f ≡ f′. This result improves a result due to Li and Yi.

Date: 2009
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https://doi.org/10.1155/2009/847690

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