Address: P.G. Department of Mathematics, University of Kashmir, Hazratbal, Srinagar-190006, India
E-mail: dr.narather@gmail.com
Abstract: Let $P(z)=\sum _{\nu =0}^{n}a_{\nu }z^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $|z|<1$, then for $1\le p<\infty $ and $R>1$, Boas and Rahman proved \[\left\Vert P(Rz)\right\Vert _{p}\le \big (\left\VertR^{n}+z\right\Vert_{p}/\left\Vert1+z\right\Vert_{p}\big )\left\Vert P\right\Vert _{p}.\] In this paper, we improve the above inequality for $0\le p < \infty $ by involving some of the coefficients of the polynomial $P(z)$. Analogous result for the class of polynomials $P(z)$ having no zero in $|z|>1$ is also given.
AMSclassification: primary 26D10; secondary 41A17, 30C15.
Keywords: polynomials, integral inequalities, complex domain.
DOI: 10.5817/AM2022-3-159