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Volume 32, Issue 2
On Growth of Polynomials with Restricted Zeros

Abdullah Mir & G. N. Parrey

Anal. Theory Appl., 32 (2016), pp. 181-188.

Published online: 2016-04

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  • Abstract

Let $P(z)$ be a polynomial of degree $n$ which does not vanish in $|z|< k $, $k\geq 1$. It is known that for each $0\leq s< n$ and $1\leq R\leq k$, $$M\big(P^{(s)},R\big)\leq \Big(\frac{1}{R^{s}+k^{s}}\Big)\Big[\Big\{\frac{d^{(s)}}{dx^{(s)}}(1+x^{n})\Big\}_{x=1}\Big]\Big(\frac{R+k}{1+k}\Big)^{n}M(P,1).$$ In this paper, we obtain certain extensions and refinements of this inequality by involving binomial coefficients and some of the coefficients of the polynomial $P(z)$.

  • AMS Subject Headings

30A10, 30C10, 30C15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mabdullah mir@yahoo.co.in ( Abdullah Mir)

  • BibTex
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  • TXT
@Article{ATA-32-181, author = {Abdullah Mir , and G. N. Parrey , }, title = {On Growth of Polynomials with Restricted Zeros}, journal = {Analysis in Theory and Applications}, year = {2016}, volume = {32}, number = {2}, pages = {181--188}, abstract = {

Let $P(z)$ be a polynomial of degree $n$ which does not vanish in $|z|< k $, $k\geq 1$. It is known that for each $0\leq s< n$ and $1\leq R\leq k$, $$M\big(P^{(s)},R\big)\leq \Big(\frac{1}{R^{s}+k^{s}}\Big)\Big[\Big\{\frac{d^{(s)}}{dx^{(s)}}(1+x^{n})\Big\}_{x=1}\Big]\Big(\frac{R+k}{1+k}\Big)^{n}M(P,1).$$ In this paper, we obtain certain extensions and refinements of this inequality by involving binomial coefficients and some of the coefficients of the polynomial $P(z)$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2016.v32.n2.7}, url = {http://global-sci.org/intro/article_detail/ata/4664.html} }
TY - JOUR T1 - On Growth of Polynomials with Restricted Zeros AU - Abdullah Mir , AU - G. N. Parrey , JO - Analysis in Theory and Applications VL - 2 SP - 181 EP - 188 PY - 2016 DA - 2016/04 SN - 32 DO - http://doi.org/10.4208/ata.2016.v32.n2.7 UR - https://global-sci.org/intro/article_detail/ata/4664.html KW - Polynomial, maximum modulus principle, zeros. AB -

Let $P(z)$ be a polynomial of degree $n$ which does not vanish in $|z|< k $, $k\geq 1$. It is known that for each $0\leq s< n$ and $1\leq R\leq k$, $$M\big(P^{(s)},R\big)\leq \Big(\frac{1}{R^{s}+k^{s}}\Big)\Big[\Big\{\frac{d^{(s)}}{dx^{(s)}}(1+x^{n})\Big\}_{x=1}\Big]\Big(\frac{R+k}{1+k}\Big)^{n}M(P,1).$$ In this paper, we obtain certain extensions and refinements of this inequality by involving binomial coefficients and some of the coefficients of the polynomial $P(z)$.

Abdullah Mir , and G. N. Parrey , . (2016). On Growth of Polynomials with Restricted Zeros. Analysis in Theory and Applications. 32 (2). 181-188. doi:10.4208/ata.2016.v32.n2.7
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