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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 $r$-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
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 $L ^1$-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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