Entire functions that have the same fixed points with their first derivatives

Let <f>f</f> be a nonconstant entire function, and let <f>k</f> <f>(&ges;2)</f> be an integer. We denote by <f>Ef(z)={z∈C: f(z)=z, counting multiplicities}</f> the set consisting of all the fixed points of <f>f</f>. This paper proves that if <f>f</f> and <f>f′</f> have the same fixed points, namely, <f>Ef(z)=Ef′(z)</f>, and if <f>f(k)(z)=z</f> whenever <f>f(z)=z</f>, then <f>f≡f′</f>. [Copyright &y& Elsevier]