TY - JOUR
T1 - Some Results Concerning Growth of Polynomials
JO - Analysis in Theory and Applications
VL - 1
SP - 37
EP - 46
PY - 2013
DA - 2013/03
SN - 29
DO - http://doi.org/10.4208/ata.2013.v29.n1.5
UR - https://global-sci.org/intro/article_detail/ata/4513.html
KW - Polynomial, inequality, maximum modulus, growth of polynomial.
AB - <p style="text-align: justify;">Let $P(z)$ be a polynomial of degree $n$ having no zeros in $|z|&lt; 1$, then for every real or complex number $\beta$ with $|\beta|\leq 1$, and $|z|=1$, $R\geq 1$, it is proved by Dewan et al. [4] that$$\Big|P(Rz)+\beta\Big(\frac{R+1}{2}\Big)^n P(z)\Big|\leq\frac{1}{2}\Big\{\Big(\Big|R^n+\beta\Big(\frac{R+1}{2}\Big)^n\Big|+\Big|1+\beta\Big(\frac{R+1}{2}\Big)^n\Big|\Big)\max_{|z|=1}|P(z)|$$ $$-\Big(\Big|R^n+\beta\Big(\frac{R+1}{2}\Big)^n\Big|-\Big|1+\beta\Big(\frac{R+1}{2}\Big)^n\Big|\Big)\min_{|z|=1}|P(z)|\Big\}.$$ In this paper we generalize the above inequality for polynomials having no zeros in $|z| &lt; k$, $k\leq 1$. Our results generalize certain well-known polynomial inequalities.</p>