Some Identities and Inequalities for Derivatives Original Research Article

We introduce some identities for the derivative of a trigonometric polynomial which are obtained from the identity of Riesz. We then use these new identities to derive some inequalities for derivatives of trigonometric and algebraic polynomials. Among our results are a weighted Lp inequality relating the derivative of a trigonometric polynomial to its Lp modulus and simple proofs for the inequalities of Brudnyi and Dzyadyk. We are able to give values to the constants in these inequalities.