arrow
Volume 29, Issue 1
Some Results Concerning Growth of Polynomials

A. Zireh, E. Khojastehnejhad & S. R. Musawi

Anal. Theory Appl., 29 (2013), pp. 37-46.

Published online: 2013-03

Export citation
  • Abstract

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.

  • AMS Subject Headings

30A10, 30C10, 30E15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{ATA-29-37, author = {}, title = {Some Results Concerning Growth of Polynomials}, journal = {Analysis in Theory and Applications}, year = {2013}, volume = {29}, number = {1}, pages = {37--46}, abstract = {

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} }
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 -

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.

A. Zireh, E. Khojastehnejhad & S. R. Musawi. (1970). Some Results Concerning Growth of Polynomials. Analysis in Theory and Applications. 29 (1). 37-46. doi:10.4208/ata.2013.v29.n1.5
Copy to clipboard
The citation has been copied to your clipboard