Lower and upper bounds of integral mean estimate for polar derivative of a polynomial

Abstract

If p(z) is a polynomial of degree n having all its zeros in $|z|\leq k$, $k\geq1$, then for $r\geq 1$, Aziz [J. Approx. Theory, 55 (1988), 232-239] proved

\begin{equation*}

\left\lbrace\int\limits_{0}^{2\pi}|1+k^ne^{i\theta}|^{r}d\theta\right\rbrace^{\frac{1}{r}}\max_{|z|=1}|p'(z)|\geq n\left\lbrace\int\limits_{0}^{2\pi}|p(e^{i\theta})|^{r}d\theta\right\rbrace^{\frac{1}{r}},
\end{equation*}

whereas, Devi et al. [Note Mat., 41 (2021), 19{29] proved that if p(z) is a polynomial of degree n having no zero in |z| < k; k 1, then for r > 0,

\begin{equation*}
k^{n}n\left\lbrace\int\limits_{0}^{2\pi}|p(e^{i\theta})|^{r}d\theta\right\rbrace^{\frac{1}{r}} \leq \left\lbrace\int\limits_{0}^{2\pi}|e^{i\theta}+k^n|^{r}d\theta\right\rbrace^{\frac{1}{r}}\left\{n\max_{|z|=1}|p(z)|-\max_{|z|=1}|p'(z)|\right\},
\end{equation*}

provided jp0(z)j and jq0(z)j attain their maxima at the same point on jzj = 1, where $q(z)=z^{n}\overline{p\left(\frac{1}{\bar z}\right)}$.

In this paper, we not only obtain improved extensions of the above inequalities into polar derivative by involving the leading coecient and the constant term of the polynomial but also give generalized integral extension of inequalities on polar derivative recently proved by Singh et al. [Complex Anal. Synerg., 9:3 (2023), 1-8].