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REFINEMENT OF TURÁN-TYPE INEQUALITY FOR A POLYNOMIAL
Keywords:
polynomials, inequalities, zeros, polar derivative.DOI:
https://doi.org/10.17654/0972087123014Abstract
Let $p(z)$ be a polynomial of degree $n$. Then the polar derivative of $p(z)$ with respect to a real or complex number $\alpha$ is defined by $D_\alpha p(z)=n p(z)+(\alpha-z) p^{\prime}(z)$. Govil and Mctume [14] proved that if $p(z)$ is a polynomial of degree $n$ having all its zeros in $|z| \leq k, k \geq 1$, then for a complex number $\alpha$ with $|\alpha| \geq 1+k+k^n$,
$$
\begin{aligned}
\max _{|z|=1}\left|D_\alpha p(z)\right| \geq & n\left(\frac{|\alpha|-k}{1+k^n}\right) \max _{|z|=1}|p(z)| \\
& +n\left(\frac{|\alpha|-\left(1+k+k^n\right)}{1+k^n}\right) \min _{|z|=k}|p(z)| .
\end{aligned}
$$
In this paper, we obtain a refinement of the above inequality.
Received: March 16, 2023
Accepted: July 4, 2023
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