On a Bojanic-stanojevic Type Inequality and Its Applications

An extension of the Bojanić–Stanojević type inequality [1] is made by considering the r-th derivate of the Dirichlet kernel D k instead of Dk. Namely, the following inequality is proved ∥∥∥∥ n ∑ k=1 αkD (r) k (x) ∥∥∥∥ 1 ≤Mpn ( 1 n n ∑ k=1 |αk| )1/p , where ‖ · ‖1 is the L-norm, {αk} is a sequence of real numbers, 1 < p ≤ 2, r = 0, 1, 2, . . . and Mp is an absolute constant dependent only on p. As an application of this inequality, it is shown that the class Fpr is a subclass of BV ∩ Cr, where Fpr is the extension of the Fomin’s class, Cr is the extension of the Garrett–Stanojević class [8] and BV is the class of all null sequences of bounded variation.