Results on the number of zeros in a disk for three types of polynomials

We impose a monotonicity condition with several reversals on the moduli of the coefficients of a polynomial. We then consider three types of polynomials: (1) those satisfying the condition on all of the coefficients, (2) those satisfying the condition on the even indexed and odd indexed coefficients separately, and (3) polynomials of the form P(z) = a(0) + Sigma(n)(j=mu) a(j)z(j) where mu >= 1 with the coefficients a(mu), a(mu)+1, ..., a(n) satisfying the condition. For each type of polynomial, we give a result which puts a bound on the number of zeros in a disk (in the complex plane) centered at the origin. For each type, we give an example showing the results are best possible.