Inequalities for the Polar Derivative of a Polynomial - Ahmad Zireh (Abstract and Applied Analysis)
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.