*S*-Fold Zeros at the Origin

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*S*-Fold Zeros at the Origin}, journal = {Analysis in Theory and Applications}, year = {2016}, volume = {32}, number = {1}, pages = {27--37}, abstract = {

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 $$\Big|zp'(z)+\frac{n\beta}{2}p(z)\Big| \leq \frac{n}{2}\Big\{\Big(\Big|\frac{\beta}{2}\Big|+\Big|1+\frac{\beta}{2}\Big|\Big)\max_{|z|=1}|p(z)|-\Big(\Big|1+\frac{\beta}{2}\Big|-\Big|\frac{\beta}{2}\Big|\Big)\min_{|z|=1}|p(z)|\Big\},$$ for any $|\beta|\leq 1$ and $|z|=1$. In this paper we improve the above inequality for the polynomial which has no zeros in $|z|< k, $ $ k\geq 1$, except $s$-fold zeros at the origin. Our results generalize certain well known polynomial inequalities.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2016.v32.n1.3}, url = {http://global-sci.org/intro/article_detail/ata/4652.html} }*S*-Fold Zeros at the Origin JO - Analysis in Theory and Applications VL - 1 SP - 27 EP - 37 PY - 2016 DA - 2016/01 SN - 32 DO - http://doi.org/10.4208/ata.2016.v32.n1.3 UR - https://global-sci.org/intro/article_detail/ata/4652.html KW - Polynomial, $s$-fold zeros, inequality, maximum modulus, derivative. AB -

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 $$\Big|zp'(z)+\frac{n\beta}{2}p(z)\Big| \leq \frac{n}{2}\Big\{\Big(\Big|\frac{\beta}{2}\Big|+\Big|1+\frac{\beta}{2}\Big|\Big)\max_{|z|=1}|p(z)|-\Big(\Big|1+\frac{\beta}{2}\Big|-\Big|\frac{\beta}{2}\Big|\Big)\min_{|z|=1}|p(z)|\Big\},$$ for any $|\beta|\leq 1$ and $|z|=1$. In this paper we improve the above inequality for the polynomial which has no zeros in $|z|< k, $ $ k\geq 1$, except $s$-fold zeros at the origin. Our results generalize certain well known polynomial inequalities.

*S*-Fold Zeros at the Origin.

*Analysis in Theory and Applications*.

*32*(1). 27-37. doi:10.4208/ata.2016.v32.n1.3