On Improved Ankeny and Rivlin Type Polynomial Inequality

For a polynomial p (z) ==₀ⁿ a_ z^ of degree n, we denote M (p, R): =|ₙ|=ₑ 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 +12) \|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+12\|p\|-n2\|p\| (1-2|aₙ|\|p\|) ₁^R (r-1) r^N-1r+2|aₙ|{\|p\|}\, dr. In this paper, we sharpen the above inequality of Dalal and Govil.