We extend Aziz and Mohammad's result that the zeros, of a polynomial P (z) = n j=0 a j z j , ta j a j−1 > 0, j = 2, 3, . . . , n for certain t ( > 0), with moduli greater than t(n − 1)/n are simple, to polynomials with complex coefficients. Then we improve their result that the polynomial P (z), of degree n, with complex coefficients, does not vanish in the discfor r < a < 2, r being the greatest positive root of the equationx n − 2x n−1 + 1 = 0, and finally obtained an upper bound, for moduli of all zeros of a polynomial, (better, in many cases, than those obtainable from many other known results).2010 Mathematics Subject Classification: Primary 30C15; Secondary 30C10. Key words and phrases: simple zeros, zero free region, refinement, upper bound for moduli of all zeros.Communicated by Gradimir Milovanović.