(untitled) (Analysis in Theory and Applications, 2020)
Let p(z)=^n_v=0a_vz^v be a polynomial of degree n, M(p,R)=:|z|=R≥ 0|p(z)| and M(p,1)=:||p||. Then according to a well-known result of Ankeny and Rivlin [1], we have for R≥ 1, M(p,R)≤ (R^n+12)||p||.This inequality has been sharpened by Govil [4], who proved that for R≥ 1, M(p,R)≤ (R^n+12)||p||-n2(||p||^2-4|a_n|^2||p||){(R-1||p||)||p||+2|a_n|-ln(1+(R-1)||p||||p||+2|a_n|)}.In this paper, we sharpen the above inequality of Govil [4], which in turn sharpens the inequality of Ankeny and Rivlin [1].