Some q-inequalities for Hausdorff operators

Abstract

We calculate the sharp bounds for some q-analysis variants of Hausdorff type inequalities of the form

{\displaystyle \int \begin{array}{l}+\infty \\ \mathrm{0}\end{array}}{\left({\displaystyle \int \begin{array}{l}+\infty \\ \mathrm{0}\end{array}}{\displaystyle \frac{\varphi (t)}{t}}f{\displaystyle \frac{x}{t}}{{\displaystyle d}}_{q}t\right)}^p{{\displaystyle d}}_{q}x\le {{\displaystyle C}}_{\varphi }{\displaystyle \int \begin{array}{l}b\\ \mathrm{0}\end{array}}{f}^p(t){{\displaystyle d}}_{q}t.

As applications, we obtain several sharp q-analysis inequalities of the classical positive integral operators, including the Hardy operator and its adjoint operator, the Hilbert operator, and the Hardy-Littlewood-Pólya operator.

Keywords

Sharp constant / Hausdorff operator / Hilbert operator / q-inequality

References

[1] BangerezakoG. Variational calculus on q-nonuniform lattices. J Math Anal Appl, 2005, 306: 161–179

[2] ChenJ, FanD, LiJ. Hausdorff operators on function spaces. Chin Ann Math Ser B, 2012, 33(4): 537–556

[3] ErnstT. A new notation for q-calculus and a new q-Taylor formula. UUDM Report, 1999, 25

[4] ErnstT. A Comprehensive Treatment of q-Calculus. Basel: Birkhäuser, 2012

[5] EulerL. Methodus generalis summandi progressiones. Commentarii academiae Scientiarum Petropolitanae, 1732-1733, 6: 68–97

[6] EulerL. Inventio summae cuiusque seriei ex dato termino generali. Commentarii academiae Scientiarum Petropolitanae, 1736, 8: 9–22

[7] ExtonH. q-hypergeometric Functions and Applications. New York: Halstead Press, 1983

[8] HardyG H, LittlewoodJ E, PólyaG. Inequalities. 2nd ed. Cambridge: Cambridge Univ Press, 1952

[9] JacksonF H. On q-definite integrals. Quart J Pure Appl Math, 1910, 41: 193–203

[10] JacksonF H. On q-difference equations. Amer J Math, 1910, 32: 305–314

[11] JacobiC G J. Fundamenta Nova Theoriae Functionum Ellipticarum. Cambridge: Cambridge Univ Press, 2012 (in Latin)

[12] KacV, CheungP. Quantum Calculus. New York: Springer-Verlag, 2002

[13] LernerA K, LiflyandE. Multidimensional Hausdorff operators on the real Hardy spaces. J Aust Math Soc, 2007, 83: 79–86

[14] LiflyandE. Open problems on Hausdorff operators. In: Complex Analysis and Potential Theory, Proceedings of the Conference Satellite to ICM 2006, Istanbul, Turkey. 2006, 280–284

[15] LiflyandE, MiyachiA. Boundedness of the Hausdorff operators in Hp spaces, 0<p<1. Studia Math, 2009, 194: 279–292

[16] LiflyandE, MóreczF. The Hausdorff operator is bounded on real H1 space. Proc Amer Math Soc, 2000, 128: 1391–1396

[17] MaligrandaL, OinarovR, PerssonL E. On Hardy q-inequalities. Czechoslovak Math J, 2014, 64: 659–682

RIGHTS & PERMISSIONS

2017 Higher Education Press and Springer-Verlag Berlin Heidelberg