Abstract
If $P(z)$ is a polynomial of degree $n$ having all its zeros in $|z|\le k, k\le 1,$ Rather, Gulzar and Ahangar [8] proved that for every $\alpha \in \mathbb{C}$ with $|\alpha|\ge k$ and $\gamma >0,$ \[ n(|\alpha|-k)\Bigg\{\int_0^{2\pi}\Big|\frac{P(e^{i\theta})}{D_{\alpha}P(e^{i\theta})}\Big|^\gamma d\theta\Bigg\}^\frac{1}{\gamma}&\leq \Bigg\{\int_0^{2\pi}\Big|1+ke^{i\theta}\Big|^\gamma d\theta\Bigg\}^\frac{1}{\gamma}. \] In this paper, we shall obtain a result which generalizes and sharpens the above inequality by obtaining a bound that depends upon the location of all the zeros of $P(z)$ rather than just on the location of the zero of largest modulus.
Citation
Imtiaz Hussain. Abdullah Mir. "On some results of M.A. Malik concerning polynomials." Funct. Approx. Comment. Math. 57 (2) 143 - 149, December 2017. https://doi.org/10.7169/facm/1620
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