Let E be the class of all entire functions \(f(t)=\sum^{\infty}_{k=0}a_ kt^ k\) with \(a_ 0=1\), \(a_ k>0\) for \(k=1,2,3,...\), and \(\int^{\infty}_{0}t^ k(f(t))^{-1}dt=1/a_ k, k=0,1,2,... \). A conjecture of Renyi and Vincze is verified by proving the exponential function \(f(t)=e^ t\) is the only member of E. The proof depends heavily on the inequality of Hayman and Vincze that \(| f'(t)(f(t))^{-1}-1|0\) and all \(f\in E\).