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Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives

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  • Jianming Qi
  • Feng Lü
  • Ang Chen

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.

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Handle: RePEc:wly:jnlaaa:v:2009:y:2009:i:1:n:847690
DOI: 10.1155/2009/847690
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