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) = a0+ Σnj=µ ajzj where µ ≥ 1 with the coefficients aµ; aµ+1;…; an 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.

We put restrictions on the coefficients of polynomials and give bounds concerning the number of zeros in a specific region. The restrictions involve a monotonicity-type condition on the coefficients of the even powers of the variable and on the coefficients of the odd powers of the variable (treated separately). We present results by imposing the restrictions on the moduli of the coefficients, the real and imaginary parts of the coefficients, and the real parts (only) of the coefficients.