Volume 1, Issue 2, 2000

Article 13

http://jipam.vu.edu.au/v1n2/006_99.html

ON AN BOJANIC-STANOJEVIC TYPE INEQUALITY AND ITS APPLICATIONS

ZIVORAD TOMOVSKI
E-Mail: tomovski@iunona.pmf.ukim.edu.mk

FACULTY OF NATURAL AND MATHEMATICAL SCIENCES,
P.O. BOX 162
91000 SKOPJE, MACEDONIA

Received 22 September, 1999; accepted 7 March, 2000.
Communicated by: H. M. Srivastava

ABSTRACT. An extension of the Bojanic-Stanojevic type inequality [1] is made by considering the -th derivate of the Dirichlet's kernel $D_k^{(r)}$ instead of . Namely, the following inequality is proved:

$\begin{displaymath}\Bigl\Vert\sum_{k=1}^n\alpha_k D _k^{(r)}(x)\Bigr\Vert_1\le M... ...gl({{1}\over{n}}\sum_{k=1}^n \vert\alpha_k\vert^p\Bigr)^{1/p} ,\end{displaymath}$

where $\Vert\cdot\Vert_1$ is the

-norm, {

} is a sequence of real numbers, $1<p\le 2$ , $r= 0,1,2,\ldots$ and

is an absolute constant dependent only on

. As an application of this inequality, it is shown that the class ${\cal F}_{pr}$ is a subclass of ${\cal B}{\cal V}\cap {\cal C}_r$ , where ${\cal F}_{pr}$ is the extension of the Fomin's class, ${\cal C}_r$ is the extension of the Garrett-Stanojevic class [8] and ${\cal B}{\cal V}$ is the class of all null sequences of bounded variation.

[1] R. BOJANIC and C.V. STANOJEVIC, A class of L¹-convergence, Trans. Amer. Math. Soc., 269 (1982), 677-683.
[8] Z. TOMOVSKI, An extension of the Garrett- Stanojevic class, Approx. Theory Appl., 16(1) (2000) 46–51. [ONLINE] A corrected version is available in the RGMIA Research Report Collection, 3(4), Article 3, 2000. URL: http://rgmia.vu.edu.au/v3n4.html

Key words:
Bojanic-Stanojevic inequality, Sidon-Fomin's inequality, Bernstein's inequality, L¹-convergence, cosine series.

2000 Mathematics Subject Classification: 26D15, 42A20.

Download this article (PDF):

Suitable for a printer:

Suitable for a monitor:

To view these files we recommend you save them to your file system and then view by using the Adobe Acrobat Reader.

That is, click on the icon using the 2nd mouse button and select "Save Target As..." (Microsoft Internet Explorer) or "Save Link As..." (Netscape Navigator).

See our PDF pages for more information.

Other papers in this issue

Ostrowski type inequalities from a linear functional point of view
I. Gavrea and B. Gavrea

A Grüss type inequality for sequences of vectors in inner product spaces and applications
S.S. Dragomir

On an Bojanic-Stanojevic type inequality
Z. Tomovski

Regularity results for vector fields of bounded distortion and applications
A. Fiorenza and F. Giannetti

Inequalities for power-exponential functions
F. Qi and L. Debnath

On the generalized strongly nonlinear implicit quasivariational inequalities for set-valued mappings
Y.J. Cho, Z. He, Y. F. Cao and N. J. Huan

A new look at Newton's inequalities
C. P. Niculescu

An application of almost increasing and d-quasi monotone sequences
H. Bor

Several integral inequalities
F. Qi

On integral inequalities of Gronwall-Bellman-Bihari type in several variables
J. A. Oguntuase

Some inequalities for the expectation and variance of a random variable whose PDF is n-time differentiable
N.S. Barnett, P. Cerone, S.S. Dragomir and J. Roumeliotis

On a Strengthened Hardy-Hilbert Inequality
B. Yang

Previous issues

Volume 1, Issue 1, 2000