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Let $P(z)$ be a polynomial of degree $n$ having no zeros in $|z|< 1$, 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 $|z| < k$, $k\leq 1$. Our results generalize certain well-known polynomial inequalities.
}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2013.v29.n1.5}, url = {http://global-sci.org/intro/article_detail/ata/4513.html} }Let $P(z)$ be a polynomial of degree $n$ having no zeros in $|z|< 1$, 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 $|z| < k$, $k\leq 1$. Our results generalize certain well-known polynomial inequalities.