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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| Format: | Article |
| Language: | English |
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Hindawi Limited
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/181934 |