Generalization of a Result Involving Maximum Moduli of Self-Inversive Polynomial and its Derivative

For an arbitrary polynomial p(z), let M(p, r)=max_|z|=_r|p(z)|. For a self-inversive polynomial p(z) (with respect to the unit circle), of degree n it is known that

M(p,1)=n/₂ M(p, 1).

By considering a self-inversive polynomial p(z) (with respect to the circle |z|=k, (k>0)), of degree n we have obtained

M(p',1)≥{ n/(1+kⁿ)} M(p,1), k≥1, {n/(1 + k²⁻ⁿ)}M(p,1), k < 1

M(p',1) {n/(1+ k²⁻ⁿ)} M(p,1), k≥1; {n/(1+kⁿ)}M(p,1), k < 1.

( provided p′(z) and p′(k²z) attain their maximum moduli on the unit circle, at the same point ) ; a generalization of this result.