Volume 30, Issue 3
Some Generalization for an Operator Which Preserving Inequalities Between Polynomials

Anal. Theory Appl., 30 (2014), pp. 260-274.

Published online: 2014-10

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• Abstract

For a polynomial $p(z)$ 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), 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 complex number $\beta$ with $|\beta|\leq 1$ and $|z|=1$. In this paper we consider the operator $B$, which carries a polynomial $p(z)$ into$$B[p(z)]:=\lambda_0p(z)+\lambda_1\Big(\frac{nz}{2}\Big)\frac{p'(z)}{1!}+\lambda_2\Big(\frac{nz}{2}\Big)^2\frac{p''(z)}{2!},$$ where $\lambda_0,$ $\lambda_1$, and $\lambda_2$ are such that all the zeros of $u(z)=\lambda_0+c(n,1)\lambda_1z+c(n,2)\lambda_2z^2$ lie in the half plane $|z|\leq |z-{n}/{2}|.$ By using the operator $B$, we present a generalization of result of Dewan. Our result generalizes certain well-known polynomial inequalities.

• AMS Subject Headings

30A10, 30C10, 30D15

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@Article{ATA-30-260, author = {}, title = {Some Generalization for an Operator Which Preserving Inequalities Between Polynomials}, journal = {Analysis in Theory and Applications}, year = {2014}, volume = {30}, number = {3}, pages = {260--274}, abstract = {

For a polynomial $p(z)$ 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), 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 complex number $\beta$ with $|\beta|\leq 1$ and $|z|=1$. In this paper we consider the operator $B$, which carries a polynomial $p(z)$ into$$B[p(z)]:=\lambda_0p(z)+\lambda_1\Big(\frac{nz}{2}\Big)\frac{p'(z)}{1!}+\lambda_2\Big(\frac{nz}{2}\Big)^2\frac{p''(z)}{2!},$$ where $\lambda_0,$ $\lambda_1$, and $\lambda_2$ are such that all the zeros of $u(z)=\lambda_0+c(n,1)\lambda_1z+c(n,2)\lambda_2z^2$ lie in the half plane $|z|\leq |z-{n}/{2}|.$ By using the operator $B$, we present a generalization of result of Dewan. Our result generalizes certain well-known polynomial inequalities.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2014.v30.n3.2}, url = {http://global-sci.org/intro/article_detail/ata/4490.html} }
TY - JOUR T1 - Some Generalization for an Operator Which Preserving Inequalities Between Polynomials JO - Analysis in Theory and Applications VL - 3 SP - 260 EP - 274 PY - 2014 DA - 2014/10 SN - 30 DO - http://doi.org/10.4208/ata.2014.v30.n3.2 UR - https://global-sci.org/intro/article_detail/ata/4490.html KW - $B$-operator, inequality, polynomial, maximum modulus, restricted zeros. AB -

For a polynomial $p(z)$ 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), 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 complex number $\beta$ with $|\beta|\leq 1$ and $|z|=1$. In this paper we consider the operator $B$, which carries a polynomial $p(z)$ into$$B[p(z)]:=\lambda_0p(z)+\lambda_1\Big(\frac{nz}{2}\Big)\frac{p'(z)}{1!}+\lambda_2\Big(\frac{nz}{2}\Big)^2\frac{p''(z)}{2!},$$ where $\lambda_0,$ $\lambda_1$, and $\lambda_2$ are such that all the zeros of $u(z)=\lambda_0+c(n,1)\lambda_1z+c(n,2)\lambda_2z^2$ lie in the half plane $|z|\leq |z-{n}/{2}|.$ By using the operator $B$, we present a generalization of result of Dewan. Our result generalizes certain well-known polynomial inequalities.

A. Zireh, S. Kumar & K. K. Dewan. (1970). Some Generalization for an Operator Which Preserving Inequalities Between Polynomials. Analysis in Theory and Applications. 30 (3). 260-274. doi:10.4208/ata.2014.v30.n3.2
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