On Improved Ankeny and Rivlin Type Polynomial Inequality

Abstract

For a polynomial $$p(z)=\sum_{\nu=0}^n a_\nu z^\nu$$ of degree $$n$$, we denote $$M(p,R):=\max_{|z|=R \geq 0} |p(z)|$$ and $$M(p,1):=\|p\|$$. A well-known result of Ankeny and Rivlin [Pacific J. Math., 5(2)(1955), 849–862] states that if $$p(z)\neq 0$$ in $$|z|<1$$ and $$R\geq 1$$, then $$M(p, R)\le\left(\frac{R^n +1}{2}\right)\|p\|$$.

This inequality has been sharpened by Dalal and Govil [Anal. Theory Appl., 36(2)(2020), 225–234], who proved that for $$p(z)\neq 0$$ in $$|z|<1$$, $$R \geq 1$$ and any $$N$$, $$1 \leq N \leq n$$, $$M(p,R)\le\frac{R^n+1}{2}\|p\|-\frac{n}{2}\|p\|\left(1-\frac{2|a_n|}{\|p\|}\right) \int_{1}^{R}\frac{(r-1)r^{N-1}}{r+\frac{2|a_n|}{\|p\|}}\,dr$$.

In this paper, we sharpen the above inequality of Dalal and Govil.