Integral Analogue of Turán-type Inequalities Concerning the Polar Derivative of a Polynomial
Year: 2024
Type: article
Abstract: If \(w(\zeta)\) is a polynomial of degree $n$ with all its zeros in \(|\zeta|\leq \Delta, \Delta\geq 1\) and any real \(\gamma\geq 1\), Aziz proved the integral inequality [1] $$ \left\lbrace\int_{0}^{2\pi}\left|1+\Delta^ne^{i\theta}\right|^{\gamma}d\theta\right\rbrace^{{1}/{\gamma}}\max_{|\zeta|=1}|w^{\prime}(\zeta)|\geq n\left\lbrace\int_{0}^{2\pi}\left|w\left(e^{i\theta}\right)\right|^{\gamma}d\theta\right\rbrace^{{1}/{\gamma}}.$$ In this article, we establish a refined extension of the above integral inequality by using the polar derivative instead of the ordinary derivative consisting of the leading coefficient and the constant term of the polynomial. Besides, our result also yields other intriguing inequalities as special cases.
Source: Ural mathematical journal
Institution: National Institute of Technology Manipur
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4405869504