Inequalities for the Polar Derivative of a Polynomial
For a polynomial 𝑝(𝑧) of degree 𝑛, we consider an operator 𝐷𝛼 which map a polynomial 𝑝(𝑧) into 𝐷𝛼𝑝(𝑧)∶=(𝛼−𝑧)𝑝′(𝑧)+𝑛𝑝(𝑧) with respect to 𝛼. It was proved by Liman et al. (2010) that if 𝑝(𝑧) has no zeros in |𝑧|<1, then for all 𝛼,𝛽∈ℂ with |𝛼|≥1,|𝛽|≤1 and |𝑧|=1, |𝑧𝐷𝛼𝑝(𝑧)+𝑛𝛽((|𝛼|−1)/2)𝑝(𝑧)|≤(𝑛/2){[|𝛼+...
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/181934 |
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| author | Ahmad Zireh |
| author_facet | Ahmad Zireh |
| author_sort | Ahmad Zireh |
| collection | DOAJ |
| description | For a polynomial 𝑝(𝑧) of degree 𝑛, we consider an operator 𝐷𝛼 which map a polynomial 𝑝(𝑧) into 𝐷𝛼𝑝(𝑧)∶=(𝛼−𝑧)𝑝′(𝑧)+𝑛𝑝(𝑧) with respect to 𝛼. It was proved by Liman et al. (2010) that if 𝑝(𝑧) has no zeros in |𝑧|<1, then for all 𝛼,𝛽∈ℂ with |𝛼|≥1,|𝛽|≤1 and |𝑧|=1, |𝑧𝐷𝛼𝑝(𝑧)+𝑛𝛽((|𝛼|−1)/2)𝑝(𝑧)|≤(𝑛/2){[|𝛼+𝛽((|𝛼|−1)/2)|+|𝑧+𝛽((|𝛼|−1)/2)|]max|𝑧|=1|𝑝(𝑧)|−[|𝛼+𝛽((|𝛼|−1)/2)|−|𝑧+𝛽((|𝛼|−1)/2)|]min|𝑧|=1|𝑝(𝑧)|}. In this paper we extend the above inequality for the polynomials having no zeros in |𝑧|<𝑘, where 𝑘≤1. Our result generalizes certain well-known polynomial inequalities. |
| format | Article |
| id | doaj-art-f3d7e6f7324440c2aed8d535db84fc8f |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
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| series | Abstract and Applied Analysis |
| spelling | doaj-art-f3d7e6f7324440c2aed8d535db84fc8f2025-02-03T06:08:20ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/181934181934Inequalities for the Polar Derivative of a PolynomialAhmad Zireh0Department of Mathematics, Shahrood University of Technology, P.O. Box 316-36155, Shahrood, IranFor a polynomial 𝑝(𝑧) of degree 𝑛, we consider an operator 𝐷𝛼 which map a polynomial 𝑝(𝑧) into 𝐷𝛼𝑝(𝑧)∶=(𝛼−𝑧)𝑝′(𝑧)+𝑛𝑝(𝑧) with respect to 𝛼. It was proved by Liman et al. (2010) that if 𝑝(𝑧) has no zeros in |𝑧|<1, then for all 𝛼,𝛽∈ℂ with |𝛼|≥1,|𝛽|≤1 and |𝑧|=1, |𝑧𝐷𝛼𝑝(𝑧)+𝑛𝛽((|𝛼|−1)/2)𝑝(𝑧)|≤(𝑛/2){[|𝛼+𝛽((|𝛼|−1)/2)|+|𝑧+𝛽((|𝛼|−1)/2)|]max|𝑧|=1|𝑝(𝑧)|−[|𝛼+𝛽((|𝛼|−1)/2)|−|𝑧+𝛽((|𝛼|−1)/2)|]min|𝑧|=1|𝑝(𝑧)|}. In this paper we extend the above inequality for the polynomials having no zeros in |𝑧|<𝑘, where 𝑘≤1. Our result generalizes certain well-known polynomial inequalities.http://dx.doi.org/10.1155/2012/181934 |
| spellingShingle | Ahmad Zireh Inequalities for the Polar Derivative of a Polynomial Abstract and Applied Analysis |
| title | Inequalities for the Polar Derivative of a Polynomial |
| title_full | Inequalities for the Polar Derivative of a Polynomial |
| title_fullStr | Inequalities for the Polar Derivative of a Polynomial |
| title_full_unstemmed | Inequalities for the Polar Derivative of a Polynomial |
| title_short | Inequalities for the Polar Derivative of a Polynomial |
| title_sort | inequalities for the polar derivative of a polynomial |
| url | http://dx.doi.org/10.1155/2012/181934 |
| work_keys_str_mv | AT ahmadzireh inequalitiesforthepolarderivativeofapolynomial |