On Improved Ankeny and Rivlin Type Polynomial Inequality

For a polynomial p(z)=∑ν=0ⁿ a_ν z^ν of degree n, we denote M(p,R):=max|z|=R ≥ 0 |p(z)| and $$M(p,1):=\|p\|$$. A well-known result of Ankeny and Rivlin [Pacific J. Math., 5(2)(1955), 849–862] states that if p(z)≠ 0 in $$|z|<1$$ and R≥ 1, then M(p, R)≤(Rⁿ +1/2)\|p\|. This inequality has been sharpened by Dalal and Govil [Anal. Theory Appl., 36(2)(2020), 225–234], who proved that for p(z)≠ 0 in $$|z|<1$$, R ≥ 1 and any N, 1 ≤ N ≤ n, M(p,R)ⁿ+1/2\|p\|-n/2\|p\|(1-2|aₙ|/\|p\|) ∫₁R(r-1)rN-1r+2|aₙ|/\|p\|\,dr. In this paper, we sharpen the above inequality of Dalal and Govil.