Generalization of the fricke theorem on entire functions of finite index

We prove that, for every sequence (a _k) of complex numbers satisfying the conditions Σ(1/|a _k |) < ∞ and |a _k+1| − |a _k | ↗ ∞ (k → ∞), there exists a continuous functionl decreasing to 0 on 0, + ∞] and such thatf(z) = Π(1 −z/|a _k |) is an entire function of finitel-index.