On certain new characterizations of weights in higher order Wirtinger-type inequalities via time scales calculus
-
Mahmoud M. Osman
,
Mario Krnić
,
Ravi P. Agarwal
and
Samir H. Saker
Abstract
In this paper, we establish a new characterization of weights in dynamic inequality of Wirtinger’s type involving higher order delta derivatives on time scales. More precisely, conditions on nonnegative rd-continuous weighted functions u and υ are given which ensure that a mixed norm dynamic inequality of the form
holds for
Acknowledgements
The authors would like to thank the anonymous referee for some valuable comments and useful suggestions.
References
[1] R. Adegoke and J. A. Oguntuase, On weighted inequality of Hardy type for higher order derivatives, Kragujevac J. Math. 23 (2001), 45–51. Search in Google Scholar
[2] R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999), no. 1–2, 3–22. 10.1007/BF03322019Search in Google Scholar
[3] R. P. Agarwal, D. O’Regan and S. H. Saker, Dynamic Inequalities on Time Scales, Springer, Cham, 2014. 10.1007/978-3-319-11002-8Search in Google Scholar
[4] R. P. Agarwal, D. O’Regan and S. H. Saker, Hardy Type Inequalities on Time Scales, Springer, Cham, 2016. 10.1007/978-3-319-44299-0Search in Google Scholar
[5] R. P. Agarwal and P. Y. H. Pang, Remarks on the generalizations of Opial’s inequality, J. Math. Anal. Appl. 190 (1995), no. 2, 559–577. 10.1006/jmaa.1995.1091Search in Google Scholar
[6] K. F. Andersen and H. P. Heinig, Weighted norm inequalities for certain integral operators, SIAM J. Math. Anal. 14 (1983), no. 4, 834–844. 10.1137/0514064Search in Google Scholar
[7] P. R. Beesack and H. P. Heinig, Hardy’s inequalities with indices less than 1, Proc. Amer. Math. Soc. 83 (1981), no. 3, 532–536. 10.2307/2044112Search in Google Scholar
[8] R. Bibi, M. Bohner, J. Pec̆arić and S. Varošanec, Minkowski and Beckenbach–Dresher inequalities and functionals on time scales, J. Math. Inequal. 7 (2013), no. 3, 299–312. 10.7153/jmi-07-28Search in Google Scholar
[9] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, 2001. 10.1007/978-1-4612-0201-1Search in Google Scholar
[10] M. J. Bohner, M. M. Osman and S. H. Saker, General higher-order dynamic Opial inequalities with applications, Dynam. Systems Appl. 26 (2017), no. 1, 65–79. Search in Google Scholar
[11] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), no. 4, 405–408. 10.4153/CMB-1978-071-7Search in Google Scholar
[12] I. Brnetić and J. Pec̆arić, Some new Opial-type inequalities, Math. Inequal. Appl. 1 (1998), no. 3, 385–390. 10.7153/mia-01-37Search in Google Scholar
[13] R. R. Hall, The behaviour of the Riemann zeta-function on the critical line, Mathematika 46 (1999), no. 2, 281–313. 10.1112/S0025579300007762Search in Google Scholar
[14] R. R. Hall, A Wirtinger type inequality and the spacing of the zeros of the Riemann zeta-function, J. Number Theory 93 (2002), no. 2, 235–245. 10.1006/jnth.2001.2719Search in Google Scholar
[15] R. R. Hall, Generalized Wirtinger inequalities, random matrix theory, and the zeros of the Riemann zeta-function, J. Number Theory 97 (2002), no. 2, 397–409. 10.1016/S0022-314X(02)00005-7Search in Google Scholar
[16] G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), no. 3–4, 314–317. 10.1007/BF01199965Search in Google Scholar
[17] G. H. Hardy, Notes on some points in the integral calculus (LX): An inequality between integrals, Messenger Math. 54 (1925), 150–156. Search in Google Scholar
[18] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed., Cambridge University, Cambridge, 1952. Search in Google Scholar
[19] H. P. Heinig, Variations of Hardy’s inequality, Real Anal. Exchange 5 (1979/80), no. 1, 61–81. 10.2307/44151489Search in Google Scholar
[20]
R. J. Higgins and A. Peterson,
Cauchy functions and Taylor’s formula for time scales
[21] R. Hilscher, A time scales version of a Wirtinger-type inequality and applications, J. Comput. Appl. Math. 141 (2002), 219–226. 10.1016/S0377-0427(01)00447-2Search in Google Scholar
[22] D. B. Hinton and R. T. Lewis, Discrete spectra criteria for singular differential operators with middle terms, Math. Proc. Cambridge Philos. Soc. 77 (1975), 337–347. 10.1017/S0305004100051161Search in Google Scholar
[23] C. O. Imoru, Weighted norm inequalities of Opial and Hardy-type for higher order derivatives, preprints IC/90/461, International Centre for Theoretical Physics, 1990. Search in Google Scholar
[24] A. Kufner, Higher order Hardy inequalities, Collect. Math. 44 (1993), 147–154. Search in Google Scholar
[25] A. Kufner and G. P. Kheĭnig, The Hardy inequality for higher-order derivatives (in Russian), Trudy Mat. Inst. Steklov. 192 (1990), 105–113; translation in Proc. Steklov Inst. Math. 1992 (1992), no. 3, 113–121. Search in Google Scholar
[26] B. Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. 10.4064/sm-44-1-31-38Search in Google Scholar
[27] M. M. Osman, N. Krnić and S. H. Saker, New characterizations of weights in dynamic inequalities of Hardy’s type on time scales, Turkish J. Math. (2025), to appear. Search in Google Scholar
[28] M. M. Osman, S. H. Saker and D. R. Anderson, Two weighted higher-order dynamic inequalities of Opial type with two functions, Qual. Theory Dyn. Syst. 21 (2022), no. 3, Paper No. 57. 10.1007/s12346-022-00592-zSearch in Google Scholar
[29] S. Peña, Discrete spectra criteria for singular difference operators, Math. Bohem. 124 (1999), no. 1, 35–44. 10.21136/MB.1999.125980Search in Google Scholar
[30] S. H. Saker, R. R. Mahmoud and K. R. Abdo, Characterizations of weighted dynamic Hardy-type inequalities with higher-order derivatives, J. Inequal. Appl. 2021 (2021), Paper No. 99. 10.1186/s13660-021-02633-8Search in Google Scholar
[31] S. H. Saker, M. M. Osman and D. R. Anderson, On a new class of dynamic Hardy-type inequalities and some related generalizations, Aequationes Math. 96 (2022), no. 4, 773–793. 10.1007/s00010-021-00831-9Search in Google Scholar
[32] S. H. Saker, M. M. Osman and D. R. Anderson, Two weighted norm dynamic inequalities with applications on second order half-linear dynamic equations, Qual. Theory Dyn. Syst. 21 (2022), no. 1, Paper No. 4. 10.1007/s12346-021-00534-1Search in Google Scholar
[33] S. H. Saker, M. M. Osman and M. Krnić, Higher integrability theorems on time scales from reverse Hölder’s inequalities, Appl. Anal. Discrete Math. 13 (2019), no. 3, 819–838. 10.2298/AADM181208035SSearch in Google Scholar
[34] S. H. Saker, M. M. Osman, D. O’Regan and R. P. Agarwal, Characterizations of reverse dynamic weighted Hardy-type inequalities with kernels on time scales, Aequationes Math. 95 (2021), no. 1, 125–146. 10.1007/s00010-020-00759-6Search in Google Scholar
[35] S. H. Saker, M. M. Osman, D. O’Regan and R. P. Agarwal, Hardy-type operators with general kernels and characterizations of dynamic weighted inequalities, Ann. Polon. Math. 126 (2021), no. 1, 55–78. 10.4064/ap191222-23-6Search in Google Scholar
[36]
G. Sinnamon and V. D. Stepanov,
The weighted Hardy inequality: New proofs and the case
[37] G. J. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), no. 2, 434–445. 10.1016/0022-247X(91)90316-RSearch in Google Scholar
[38] V. D. Stepanov, A weighted inequality of Hardy type for higher-order derivatives (in Russian), Trudy Mat. Inst. Steklov. 187 (1989), 178–190; translation in Proc. Steklov Inst. Math. 1990 (1990), no. 3, 205–220. Search in Google Scholar
[39] G. Talenti, Osservazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. Fis. Milano 39 (1969), 171–185. 10.1007/BF02924135Search in Google Scholar
[40] G. Tomaselli, A class of inequalities, Boll. Un. Mat. Ital. (4) 2 (1969), 622–631. Search in Google Scholar
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