Growth of the Maximum Modulus of Polynomials With Prescribed Zeros

If $p(z)$ be a polynomial of degree n which does not vanish in the disk |z|<k, then for $k=1$, it iswell known that{align*}max|z|=r<1|p(z)|&≥(r+1/2)^nmax|z|=1|p(z)|,\\{and} max|z|=R>1|p(z)|&^n+1/2max|z|=1|p(z)|.{align*}In this paper, we consider a class of lacunary polynomials and present certain generalizations as well as improvements of the above inequalities for the two cases k≥ 1 and $k<1$. 2000 Mathematics Subject Classification. 30A10, 30C10, 30C15