Turan's Inequalities for Trigonometric Polynomials

We present here a technique for establishing inequalities of the form c ‖ f ‖ ⩽ ∫ 0 2 πφ ( |f ( k )( t ) | ) d t ⩽ M ‖ f ‖ in the set of all trigonometric polynomials of order n which have only real zeros. The function φ is assumed to be convex and increasing on [0, ∞). As a corollary of the main result we get Turan's inequalities ‖ f ( k )‖ q ⩾ c ( n , k , q ) ‖ f ‖with the exact constant c ( n , k , q ) for each 1 ⩽ q ⩽ ∞, n and k .