Address: Markusa st. 22, Vladikavkaz,
362027, RNO-A, Russia
Phone: (8672)50-18-06
E-mail: rio@smath.ru
DOI: 10.23671/VNC.2016.2.5920
On a Discrete Hardy Type Inequality with Logarithmic Weight
Nasibullin R. G.
Vladikavkaz Mathematical Journal 2016. Vol. 18. Issue 2.
Abstract: We prove a new discrete Hardy type inequality with logarithmic weight. Logarithmic factors are located under the module sign. The constant in this inequality is sharp.
Keywords: Hardy type inequality, logarithmic weight, sharpness of constants.
For citation: Nasibullin R. G. On a discrete Hardy type inequality with logarithmic weight // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol.
19, no. 2, pp. 67-75.
DOI 10.23671/VNC.2016.2.5920
1. Avhadiev F. G. Neravenstva dlya integralnyh harakteristik oblastej.
Kazan', KGU, 2006. 142 p.
2. Avkhadiev F. G., Nasibullin R. G., Shafigullin I. K. Hardy-type
Inequalities with Power and Logarithmic Weights in Domains of the
Euclidean Space. Russian Mathematics (Iz. VUZ), 2011, vol. 55, no.9,
p. 76-79.
3. Nasibullin R. G. Generalizations of Hardy-type Inequalities in the
Form of Dubinskii. Mathematical Notes, 2014, vol. 95, issue 1,
p. 98-110.
4. Nasibullin R. G., Tuhvatullina A. M. Hardy Type Inequalities with
Logarithmic and Power Weights for a Special Family of Non-convex
Domains. Ufimskij matematicheskij zhurnal [Ufa Math. J.], 2013,
vol. 5, no. 2, p. 43-55 [in Russian].
5. Sobolev L. S. Izbrannye voprosy teorii funkcional'nyh prostranstv i
obobshhennyh proizvodnyh [Selected Problems in the Theory of
Function Spaces and the Generalized Derivatives]. Moscow, Nauka,
1989. 254 p.
6. Ancona A. On strong barriers and an inequality of Hardy's for
domains in Rn. J. London Math. Soc., 1986, vol. 34, p. 274-290.
7. Avkhadiev F. G. Hardy type inequalities in higher dimensions with
explicit estimate of constants. Lobachevskii J. Math., 2006,
vol. 21, p. 3-31.
8. Avkhadiev F. G., Wirths K.-J. Unified Poincare and Hardy
inequalities with sharp constants for convex domains. Z. Angew.
Math. Mech., 2007, vol. 87, p. 632-642.
9. Avkhadiev F. G., Wirths K.-J. Weighted Hardy inequalities with
sharp constants. Lobachevskii J. Math., 2010, vol. 31, p. 1-7.
10. Avkhadiev F. G., Wirths K.-J. Sharp Hardy-type inequalities with
Lamb's constants. Bull. Belg. Math. Soc. Simon Stevin, 2011, vol.
18, p. 723-736.
11. Balinsky A., Laptev A., and Sobolev A. V. Generalized Hardy
inequality for the magnetic Dirichlet forms. J. of Statistical
Physics, 2004, vol. 116, ¹ 1-4, p. 507-521.
12. Brezis H., Marcus M. Hardy's inequality revisited. Ann. Scuola
Norm. Sup. Pisa Cl. Sci. 4, 1997, vol. 25, ¹ 1-2, p. 217-237.
13. Chen C., Luor D., and Ou Z. Extensions of Hardy's inequality.
J. Math. Anal. Appl., 2002, vol. 273, p. 160-171.
14. Czmesija A. On weighted discrete Hardy's inequality for negative
power numbers. J. Math. Inequal. Appl., 2005, vol. 8, ¹ 2, p.
273-285.
15. Davies E. B. A review of Hardy's inequalities. The Maz'ya
anniversary Collection. Vol. 2. Oper. Theory Adv. Appl., 1999, Vol.
110, p. 55-67.
16. Dubinskii Yu. A. A Hardy-type inequality and its applications.
Proceedings of the Steklov Institute of Mathematics, 2010, vol. 269,
p. 106-126.
17. Gao P. Hardy type inequalities via auxiliary sequences. J. Math.
Anal. Appl., 2008, vol. 343, p. 48-57.
18. Gord Sinnamon. Weighted inequalities for positive operators.
J. Math. Inequal. Appl., 2005, vol. 8, ¹ 3, p. 419-440.
19. Hardy G. H. Note on a theorem of Hilbert. Math. Zeitschr, 1920,
vol. 6, p. 314-317.
20. Hardy G. H., Littlewood J. E., and Polya G. Inequalities.
Cambridge: Cambridge Univ. Press, 1973.
21. Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., and Laptev A. A
geometrical version of Hardy's inequality. J. Funct. Anal., 2002,
vol. 189, ¹ 2, p. 539-548.
22. Kufner A., Persson L.-E. Weighted Inequalities of Hardy's type.
World Sci. Publ. Co Inc., 2003. 376 p.
23. Landau E. A note on a theorem concerning series of positive terms:
extract from a letter of Prof. E. Landau to Prof. J. Schur. J.
London Math. Soc., 1926, vol. 1, p. 38-39.
24. Laptev A., Weidl T. Hardy inequalities for magnetic Dirichlet
forms. Operator Theory: Advances and Appl., 1999, vol. 108, p.
299-305.
25. Ling-Yau Chan. Some extensions of Hardy's inequality. Canad. Math.
Bull., 1979, vol. 22, ¹ 2, p. 165-169.
26. Liu J., Xuande Zhang, and Bo Jiang. Some generalizations and
improvements of discrete Hardy's inequality. Comp. Math. Appl.,
2012, vol. 63, p. 601-607.
27. Miclo L. An example of application of discrete Hardy's
inequalities. Markov Processes Relat. Fields, 1999, Vol. 5, ¹ 3, p.
319-330.
28. Miklyukov V. M., Vuorinen M. K. Hardy's inequalities for
\(W^{1,p}_0\) functions on Riemannian manyfolds. Proc. Amer. Math.
Soc., 1999, vol. 127, ¹ 9, p. 2745-2754.
29. Muckenhoupt B. Hardy's inequality with weights. Stud. Math., 1972,
vol. 44, ¹ 1, p. 31-38.
30. Okpoti Ch. A., Persson L.-E., and Wedestig A. Weight
characterizations for the discrete Hardy inequality with kernel.
J. Math. Inequal. Appl., 2006, vol. 2006, p. 1-14.
31. Pachpatte B. G. A note on certain inequalities related to Hardy's
inequality. Indian J. Pure Appl. Math., 1992, vol. 23, ¹ 11,
p. 773-776.
32. Pecaric J. E., Love E. R. Still more generalization of Hardy's
inequality. J. Austral. Math. Soc. Ser. A, 1995, vol. 59, p.
214-224.
33. Qiang Chen, Bicheng Yang. Half-discrete Hardy-Hilbert's inequality
with two interval variables. J. Math. Inequal. Appl., 2013, vol. 485.
34. Stepanov V. D. The weighted Hardy's inequality for nonincreasing
functions. Trans. Amer. Math. Soc., 1993, vol. 338, ¹ 1, p.
173-186.
35. Talenti G. Osservazione sopra una classe di disuguaglianze. Rend.
Semin. Mat. Efis. Milano, 1969, vol. 39, p. 171-185.
36. Tomaselli G. A class of inequalities. Boll. Unione Mat. Ital. Ser.,
1969, vol. 4, ¹ 6, p. 622-631.
37. Wannebo A. Hardy Inequalities. Proc. Amer. Math. Soc., 1990, vol.
109, ¹ 1, p. 85-95.
