Some Inequalities for the Polynomial with S-Fold Zeros at the Origin (Analysis in Theory and Applications, 2016)
Let $p(z)$ be a polynomial of degree $n$, which has no zeros in $|z|< 1$, Dewan et al. [K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 (2010), pp. 38-41] established $$\Big|zp'(z)+\frac{nβ}{2}p(z)\Big| ≤ \frac{n}{2}\Big\{\Big(\Big|\frac{β}{2}\Big|+\Big|1+\frac{β}{2}\Big|\Big)\max_{|z|=1}|p(z)|-\Big(\Big|1+\frac{β}{2}\Big|-\Big|\frac{β}{2}\Big|\Big)\min_{|z|=1}|p(z)|\Big\},$$ for any $|β|≤ 1$ and $|z|=1$. In this paper we improve the above inequality for the polynomial which has no zeros in $|z|< k, $ $ k≥ 1$, except $s$-fold zeros at the origin. Our results generalize certain well known polynomial inequalities.
Keywords: Polynomial; $s$-fold zeros; inequality; maximum modulus; derivative.