On zeros of functions from Bernstein classes

If a convergent Taylor series f(z) = Sigma(j greater than or equal to 0) a( j)z(j) satisfies the condition /a(j)/ less than or equal to M/a(k)/ for som e k and all j > k, then one can explicitly determine in terms of M and k th e radius of a centred disc containing no more man k roots of f. This proble m was solved by Yomdin and Roytwarf using the equivalence of two Bernstein classes of analytic functions and a delicate refinement of the Jensen inequ ality due to van der Poorten. We give two direct proofs of the above claim: one is more transparent though gives a slightly worse bound for the radius of the circle. The second proof generalizes the simple and popular differe ntiation-division algorithm and gives the bounds better than the original p roof. AMS classification scheme numbers: 34C05.30C15. 26C10, 34A20.