Some Results Concerning Growth of Polynomials (Analysis in Theory and Applications, 2013)
Let $P(z)$ be a polynomial of degree $n$ having no zeros in $|z|< 1$, then for every real or complex number $β$ with $|β|≤ 1$, and $|z|=1$, $R≥ 1$, it is proved by Dewan et al. [4] that$$\Big|P(Rz)+β\Big(\frac{R+1}{2}\Big)^n P(z)\Big|≤\frac{1}{2}\Big\{\Big(\Big|R^n+β\Big(\frac{R+1}{2}\Big)^n\Big|+\Big|1+β\Big(\frac{R+1}{2}\Big)^n\Big|\Big)\max_{|z|=1}|P(z)|$$ $$-\Big(\Big|R^n+β\Big(\frac{R+1}{2}\Big)^n\Big|-\Big|1+β\Big(\frac{R+1}{2}\Big)^n\Big|\Big)\min_{|z|=1}|P(z)|\Big\}.$$ In this paper we generalize the above inequality for polynomials having no zeros in $|z| < k$, $k≤ 1$. Our results generalize certain well-known polynomial inequalities.