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

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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 
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     title = {On {Hardy} $q$-inequalities},
     journal = {Czechoslovak Mathematical Journal},
     pages = {659--682},
     year = {2014},
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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

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