Volume 32, Issue 1
Some Inequalities for the Polynomial with S-Fold Zeros at the Origin

A. Zireh & M. Bidkham

Anal. Theory Appl., 32 (2016), pp. 27-37

Published online: 2016-01

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  • 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 theabove inequality for the polynomial which has no zeros in $|z|< k, $ $ k\geq 1$, except $s$-fold zeros at the origin. Our resultsgeneralize certain well known polynomial inequalities.
  • Keywords

Polynomial s-fold zeros inequality maximum modulus derivative

  • AMS Subject Headings

30A10 30C10 30D15

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COPYRIGHT: © Global Science Press

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@Article{ATA-32-27, author = {A. Zireh and M. Bidkham}, title = {Some Inequalities for the Polynomial with 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 theabove inequality for the polynomial which has no zeros in $|z|< k, $ $ k\geq 1$, except $s$-fold zeros at the origin. Our resultsgeneralize 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} }
TY - JOUR T1 - Some Inequalities for the Polynomial with S-Fold Zeros at the Origin AU - A. Zireh & M. Bidkham 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 KW - s-fold zeros KW - inequality KW - maximum modulus KW - 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 theabove inequality for the polynomial which has no zeros in $|z|< k, $ $ k\geq 1$, except $s$-fold zeros at the origin. Our resultsgeneralize certain well known polynomial inequalities.
A. Zireh & M. Bidkham. (1970). Some Inequalities for the Polynomial with S-Fold Zeros at the Origin. Analysis in Theory and Applications. 32 (1). 27-37. doi:10.4208/ata.2016.v32.n1.3
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