Geodesic
On Hardy $q$-inequalities
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 659-682
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Some $q$-analysis variants of Hardy type inequalities of the form $$ \int _0^b \bigg (x^{\alpha -1} \int _0^x t^{-\alpha } f(t) {\rm d}_q t \bigg )^{p} {\rm d}_q x \leq C \int _0^b f^p(t) {\rm d}_q t $$ with sharp constant $C$ are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.
Some $q$-analysis variants of Hardy type inequalities of the form $$ \int _0^b \bigg (x^{\alpha -1} \int _0^x t^{-\alpha } f(t) {\rm d}_q t \bigg )^{p} {\rm d}_q x \leq C \int _0^b f^p(t) {\rm d}_q t $$ with sharp constant $C$ are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.
DOI : 10.1007/s10587-014-0125-6
Classification : 26D10, 26D15, 39A13
Keywords: inequality; Hardy type inequality; Hardy operator; Riemann-Liouville operator; $q$-analysis; sharp constant; discrete Hardy type inequality 
@article{10_1007_s10587_014_0125_6,
     author = {Maligranda, Lech and Oinarov, Ryskul and Persson, Lars-Erik},
     title = {On {Hardy} $q$-inequalities},
     journal = {Czechoslovak Mathematical Journal},
     pages = {659--682},
     year = {2014},
     volume = {64},
     number = {3},
     doi = {10.1007/s10587-014-0125-6},
     mrnumber = {3298553},
     zbl = {06391518},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0125-6/}
}
TY  - JOUR
AU  - Maligranda, Lech
AU  - Oinarov, Ryskul
AU  - Persson, Lars-Erik
TI  - On Hardy $q$-inequalities
JO  - Czechoslovak Mathematical Journal
PY  - 2014
SP  - 659
EP  - 682
VL  - 64
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0125-6/
DO  - 10.1007/s10587-014-0125-6
LA  - en
ID  - 10_1007_s10587_014_0125_6
ER  - 
%0 Journal Article
%A Maligranda, Lech
%A Oinarov, Ryskul
%A Persson, Lars-Erik
%T On Hardy $q$-inequalities
%J Czechoslovak Mathematical Journal
%D 2014
%P 659-682
%V 64
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0125-6/
%R 10.1007/s10587-014-0125-6
%G en
%F 10_1007_s10587_014_0125_6
Maligranda, Lech; Oinarov, Ryskul; Persson, Lars-Erik. On Hardy $q$-inequalities. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 659-682. doi: 10.1007/s10587-014-0125-6

[1] Al-Salam, W. A.: Some fractional $q$-integrals and $q$-derivatives. Proc. Edinb. Math. Soc., II. Ser. 15 (1966), 135-140. | DOI | MR | Zbl

[2] Bangerezako, G.: Variational calculus on $q$-nonuniform lattices. J. Math. Anal. Appl. 306 (2005), 161-179. | DOI | MR | Zbl

[3] Bennett, G.: Factorizing the Classical Inequalities. Memoirs of the American Mathematical Society 576 AMS, Providence (1996). | MR | Zbl

[4] Bennett, G.: Inequalities complementary to Hardy. Q. J. Math., Oxf. II. Ser. 49 (1998), 395-432. | DOI | MR

[5] Bennett, G.: Series of positive terms. Conf. Proc. Poznań, Poland, 2003 Z. Ciesielski et al. Banach Center Publications 64 Polish Academy of Sciences, Institute of Mathematics, Warsaw (2004), 29-38. | MR | Zbl

[6] Bennett, G.: Sums of powers and the meaning of $l^p$. Houston J. Math. 32 (2006), 801-831. | MR

[7] Cass, F. P., Kratz, W.: Nörlund and weighted mean matrices as operators on $l_p$. Rocky Mt. J. Math. 20 (1990), 59-74. | DOI | MR

[8] Ernst, T.: A Comprehensive Treatment of $q$-calculus. Birkhäuser Basel (2012). | MR | Zbl

[9] Ernst, T.: The History of $q$-Calculus and a New Method. Uppsala University Uppsala (2000), http://www2.math.uu.se/research/pub/Ernst4.pdf

[10] Exton, H.: $q$-Hypergeometric Functions and Applications. Ellis Horwood Series in Mathematics and Its Applications Halsted Press, Chichester (1983). | MR | Zbl

[11] Gao, P.: A note on Hardy-type inequalities. Proc. Am. Math. Soc. 133 (2005), 1977-1984. | DOI | MR | Zbl

[12] Gao, P.: Hardy-type inequalities via auxiliary sequences. J. Math. Anal. Appl. 343 (2008), 48-57. | DOI | MR | Zbl

[13] Gao, P.: On $l^p$ norms of weighted mean matrices. Math. Z. 264 (2010), 829-848. | DOI | MR | Zbl

[14] Gao, P.: On weighted mean matrices whose $l^p$ norms are determined on decreasing sequences. Math. Inequal. Appl. 14 (2011), 373-387. | MR | Zbl

[15] Gauchman, H.: Integral inequalities in $q$-calculus. Comput. Math. Appl. 47 (2004), 281-300. | DOI | MR | Zbl

[16] Hardy, G. H., Littlewood, J. E., Pólya, G.: Inequalities. (2nd ed.), Cambridge University Press Cambridge (1952). | MR | Zbl

[17] Jackson, F. H.: On $q$-definite integrals. Quart. J. 41 (1910), 193-203.

[18] Kac, V., Cheung, P.: Quantum Calculus. Universitext Springer, New York (2002). | MR | Zbl

[19] Krasniqi, V.: Erratum: Several $q$-integral inequalities. J. Math. Inequal. 5 (2011), 451. | DOI | MR | Zbl

[20] Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy Inequality. About Its History and Some Related Results. Vydavatelský Servis Plzeň (2007). | MR | Zbl

[21] Kufner, A., Maligranda, L., Persson, L.-E.: The prehistory of the Hardy inequality. Am. Math. Mon. 113 (2006), 715-732. | DOI | MR | Zbl

[22] Kufner, A., Persson, L.-E.: Weighted Inequalities of Hardy Type. World Scientific Singapore (2003). | MR | Zbl

[23] Maligranda, L.: Why Hölder's inequality should be called Rogers' inequality. Math. Inequal. Appl. 1 (1998), 69-83. | DOI | MR | Zbl

[24] Miao, Y., Qi, F.: Several $q$-integral inequalities. J. Math. Inequal. 3 (2009), 115-121. | DOI | MR | Zbl

[25] Persson, L.-E., Samko, N.: What should have happened if Hardy had discovered this?. J. Inequal. Appl. (electronic only) 2012 (2012), Article ID 29, 11 pages. | MR | Zbl

[26] Stanković, M. S., Rajković, P. M., Marinković, S. D.: On $q$-fractional derivatives of Riemann-Liouville and Caputo type. arXiv: 0909.0387v1[math.CA], 2 Sept. 2009.

[27] Sulaiman, W. T.: New types of $q$-integral inequalities. Advances in Pure Math. 1 (2011), 77-80. | DOI | MR

Cité par Sources :