Quantitative Voronovskaya-type theorems for Fej\'er-Korovkin operators

In recent times quantitative Voronovskaya type theorems have been presented in spaces of non-periodic continuous functions. In this work we proved similar results but for Fejer-Korovkin trigonometric operators. That is we measure the rate of convergence in the associated Voronovskaya type theotem. Recall that these operators provide the optimal rate in approximation by positive linear operators. For the proofs we present new inequalities related with trigonometric polynomials as well as with the convergence factor of the Fej\'er-Korovkin operators. Our approach includes spaces of Lebesgue integrable functions.