Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T15:16:05.848Z Has data issue: false hasContentIssue false
  • English
  • Français

SHARP CONSTANTS FOR MULTIVARIATE HAUSDORFF $q$-INEQUALITIES

Part of: Inequalities Difference and functional equations Difference equations

Published online by Cambridge University Press:  07 June 2018

DASHAN FAN  and
FAYOU ZHAO
DASHAN FAN
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA Department of Mathematics, Zhejiang Normal University, Jinhua 321000, PR China email fan@uwm.edu
FAYOU ZHAO*
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, PR China email fyzhao@shu.edu.cn
*
✉fyzhao@shu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we focus on the multivariate Hausdorff operator of the form

$$\begin{eqnarray}\mathbf{H}_{\unicode[STIX]{x1D6F7}}(f)(x)=\int _{(0,+\infty )^{n}}{\displaystyle \frac{\unicode[STIX]{x1D6F7}\big(\frac{x_{1}}{t_{1}},\frac{x_{2}}{t_{2}},\ldots ,\frac{x_{n}}{t_{n}}\big)}{t_{1}t_{2}\cdots t_{n}}}f(t_{1},t_{2},\ldots ,t_{n})\,\mathbf{dt},\end{eqnarray}$$
where $\mathbf{dt}=dt_{1}\,dt_{2}\cdots \,dt_{n}$ or $\mathbf{dt}=d_{q}t_{1}\,d_{q}t_{2}\cdots d_{q}t_{n}$ is the discrete measure in $q$-analysis. The sharp bounds for the multivariate Hausdorff operator on spaces $L^{p}$ with power weights are calculated, where $p\in \mathbb{R}\backslash \{0\}$.

Keywords

sharp constantmultivariate Hausdorff operatorq-inequalities

MSC classification

Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 26D15: Inequalities for sums, series and integrals 39A13: Difference equations, scaling ($q$-differences)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 106 , Issue 2 , April 2019 , pp. 274 - 286
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The research was supported by National Natural Science Foundation of China (Grant Nos. 11471288, 11601456).

References

Anastassiou, G. A., ‘Taylor Widder representation formulae and Ostrowski, Grüss, integral means and Csiszar type inequalities’, Comput. Math. Appl. 54 (2007), 9–23.Google Scholar
Annaby, M. H. and Mansour, Z. S., q-Fractional Calculus and Equations (Springer, Heidelberg–New York, 2012).Google Scholar
Baiarystanov, A. O., Persson, L. E., Shaimardan, S. and Temirkhanova, A., ‘Some new Hardy-type inequalities in q-analysis’, J. Math. Inequal. 10 (2016), 761–781.Google Scholar
Bangerezako, G., ‘Variational calculus on q-nonuniform lattices’, J. Math. Anal. Appl. 306 (2005), 161–179.Google Scholar
Chen, J., Fan, D. and Wang, S., ‘Hausdorff operators on Euclidean space’, Appl. Math. J. Chinese Univ. Ser. B 28 (2013), 548–564.Google Scholar
Ernst, T., A Comprehensive Treatment of q-Calculus (Birkhäuser/Springer Basel AG, Basel, 2012).Google Scholar
Exton, H., q-hypergeometric Functions and Applications, Ellis Horwood Series: Mathematics and its Applications (Ellis Horwood, Chichester, UK, 1983).Google Scholar
Guo, J. and Zhao, F., ‘Some q-inequalities for Hausdorff operators’, Front. Math. China 12 (2017), 879–889.Google Scholar
Jackson, F. H., ‘On q-definite integrals’, Quart. J. Pure Appl. Math. 41 (1910), 193–203.Google Scholar
Jackson, F. H., ‘ q-difference equations’, Amer. J. Math. 32 (1910), 305–314.Google Scholar
Kac, V. and Cheung, P., Quantum Calculus (Springer-Verlag, New York, 2002).Google Scholar
Liflyand, E., ‘Hausdorff operators on Hardy spaces’, Eurasian Math. J. 4 (2013), 101–141.Google Scholar
Maligranda, L., Oinarov, R. and Persson, L. E., ‘On Hardy q-inequalities’, Czechoslovak Math. J. 64 (2014), 659–682.Google Scholar
Mitrinović, D. S., Pečarić, J. E. and Fink, A. M., Classical and New Inequalities in Analysis (Kluwer Academic, 1993).Google Scholar
Miao, Y. and Qi, F., ‘Several q-integral inequalities’, J. Math. Inequal. 3 (2009), 115–121.Google Scholar
Pachpatte, B. G., ‘On multivariable Hardy type inequalities’, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 38 (1992), 355–361.Google Scholar
Sulaiman, W. T., ‘New types of q-integral inequalities’, Adv. Pure Appl. Math. 1 (2011), 77–80.Google Scholar
Wu, X. and Chen, J., ‘Best constants for Hausdorff operators on n-dimensional product spaces’, Sci. China Math. 57 (2014), 569–578.Google Scholar
Wu, S. and Debnath, L., ‘Inequalities for convex sequences and their applications’, Comput. Math. Appl. 54 (2007), 525–534.Google Scholar