Growth of polynomials not vanishing in a disk

Abstract

This paper deals with the problem of finding some upper bound estimates for the maximal modulus of a lacunary polynomial on a disk of radius R, R≥1 under the assumption that the polynomial does not vanish in another disk with radius k, k≥1. Our results sharpen as well as generalize a result recently proved by Hussain [Indian J. Pure Appl. Math., (https://doi.org/10.1007/s13226-021-00169-7)]. Further, these results generalize as well as sharpen some known results in this direction.