## On a Bojanić-Stanojević type inequality and its applications.(English)Zbl 0970.42004

Summary: An extension of the Bojanić-Stanojević type inequality [R. Bojanić and C. V. Stanojević, Trans. Am. Math. Soc. 269, 677-683 (1982; Zbl 0493.42011)] is made by considering the $$r$$-th derivate of the Dirichlet kernel $$D_k^{(r)}$$ instead of $$D_k$$. Namely, the following inequality is proved: $\Bigl\|\sum_{k=1}^n \alpha_k D_K^{(r)} (x)\Bigr\|_1\leq M_pn^{r+1}\Bigl({1\over n}\sum^n_{k=1}|\alpha_k|^p\Bigr)^{1/p}$ where $$\|\cdot\|_1$$ is the $$L^1$$-norm, $$\{\alpha_k\}$$ is a sequence of real numbers, $$1<p\leq 2, r=0,1,2,\dots$$ and $$M_p$$ is an absolute constant dependent only on $$p$$. As an application of this inequality, it is shown that the class $$\mathcal F_{pr}$$ is a subclass of $$\mathcal B\mathcal V\cap\mathcal C_r$$, where $$\mathcal F_{pr}$$ is the extension of the Fomin’s class, $$\mathcal C_r$$ is the extension of the Garrett-Stanojević class [Ž. Tomovski, Approximation Theory Appl. 16, No. 1, 46-51 (2000; Zbl 0963.42003)] and $$\mathcal B\mathcal V$$ is the class of all null sequences of bounded variation.

### MSC:

 42A20 Convergence and absolute convergence of Fourier and trigonometric series 26D10 Inequalities involving derivatives and differential and integral operators

### Citations:

Zbl 0493.42011; Zbl 0963.42003
Full Text: