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Volume 1, Issue 2, 2000

Article 13



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

P.O. BOX 162

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 $r$-th derivate of the Dirichlet's kernel $D_k^{(r)}$ instead of $D_k$. Namely, the following inequality is proved:

\begin{displaymath}\Bigl\Vert\sum_{k=1}^n\alpha_k D _k^{(r)}(x)\Bigr\Vert_1\le
...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 $L ^1$-norm, {$a_k$} is a sequence of real numbers, $1<p\le 2$, $r= 0,1,2,\ldots$ and $M_p$ is an absolute constant dependent only on $p$. 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.

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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

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© 2000 School of Communications and Informatics, Victoria University of Technology. All rights reserved.
JIPAM is published by the School of Communications and Informatics which is part of the Faculty of Engineering and Science, located in Melbourne, Australia. All correspondence should be directed to the editorial office.


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