Comments on an $L^ 2$ inequality of A. K. Varma involving the first derivative of polynomials

Abstract:

Let ${t_n}$ be a trigonometric polynomial of degree $n$ with real coefficients, and let $w\left ( x \right ) \in {C^2}\left [ {0,\pi } \right ]$ be nonnegative. Employing a well-known result of G. Szegö, we study the extremal property of the integral \[ \int _0^\pi {{{\left ( {{{t’}_n}\left ( x \right )} \right )}^2}w\left ( x \right )dx} ,\] subject to the constraint ${\left \| {{t_n}} \right \|_\infty } \leq 1$.