INTEGRAL ANALOGUE OF TURÁN-TYPE INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIAL

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.