A generalization of Ankeny and Rivlin's result on the maximum modulus of polynomials not vanishing in the interior of the unit circle

Abstract

For an arbitrary entire function f(z), let M(f, r) = $max_{|z|}=r |f(z)|$. For a polynomial p(z) of degree n, it is known that $M(p,R)leq R^n M(p, 1)$, R>1. By considering the polynomial p(z) with no zeros in |z| 1. By considering the polynomial p(z) with no zeros in |z| < k, $(kgeq 1)$ and simultaneously thinking of $s^{th}$ derivative $(0leq s < n)$ of the polynomial, we have obtained the generalization M(p^(s),R)leq..................................... ( (1/2){ ds dRs (Rn + kn)}(2/(1 + k))nM(p, 1), R≥ k, (1/(Rs + ks))[{ ds dxs (1 + xn)}x=1]((R + k)/(1 + k))nM(p, 1), 1 ≤ R ≤ k, of Ankeny and Rivlin’s result.