Some Generalizations of Dynamic Hardy-Knopp-Type Inequalities on Time Scales

• Ahmed A. El-Deeb ^[1]
  1. [1] Al-Azhar University
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 5, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01102-z
• Enlaces
  • Texto completo
• Resumen

  • In the present paper, some new generalizations of dynamic inequalities of Hardy-type in two variables on time scales are established. The integral and discrete Hardy-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Jensen inequality and the Fubini theorem on time scales.

• Referencias bibliográficas
  • 1. Hardy, G.H.: Note on a theorem of Hilbert. Math. Z. 6(3–4), 314–317 (1920)
  • 2. Hardy, G.H.: Notes on some points in the integral calculus (lxit). Messenger Math. 57, 12–16 (1928)
  • 3. Littlewood, J.E., Hardy, G.H.: Elementary theorems concerning power series with positive coefficients and moment constants of positive...
  • 4. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Springer Science &BusinessMedia, Cambridge at the University Press, Cambridge,...
  • 5. Hardy, G.H.: Notes on some points in the integral calculus (lx). Messenger Math 54, 150–156 (1925)
  • 6. Andersen, K.F., Muckenhoupt, B.:Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions. Studia...
  • 7. Andersen, K.F., Heinig, H.P.:Weighted norm inequalities for certain integral operators. SIAM J.Math. Anal. 14(4), 834–844 (1983)
  • 8. Bennett, G.: Some elementary inequalities. Quart. J. Math. Oxford Ser. (2) 38(152), 401–425 (1987)
  • 9. Bennett, G.: Some elementary inequalities. II. Quart. J. Math. Oxford Ser. (2) 39(156), 385–400 (1988)
  • 10. Bennett, G.: Some elementary inequalities. III. Quart. J. Math.Oxford Ser. (2) 42(166), 149–174 (1991)
  • 11. Copson, E.T.: Note on Series of Positive Terms. J. London Math. Soc. 3(1), 49–51 (1928)
  • 12. Heinig, H.P.: Weighted norm inequalities for certain integral operators. II. Proc. Amer. Math. Soc. 95(3), 387–395 (1985)
  • 13. Leindler, L.: Generalization of inequalities of Hardy and Littlewood. Acta Sci. Math. (Szeged) 31, 279–285 (1970)
  • 14. Kufner, A., Persson, L.-E.:Weighted Inequalities of Hardy Type.World Scientific Publishing Co., Inc, River Edge, NJ (2003)
  • 15. Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy Inequality. Vydavatelský Servis, Plzeˇn, About its history and some related results...
  • 16. Opic, B.,Kufner, A.: Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series.Longman Scientific & Technical, Harlow (1990)
  • 17. Hilger, S.: Analysis on measure chains–a unified approach to continuous and discrete calculus. Results Math. 18(1–2), 18–56 (1990)
  • 18. Bohner, M., Peterson, A.: Advances in dynamic equations on time scales. Birkhäuser Boston Inc, Boston, MA (2003)
  • 19. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhäuser Boston, Inc., Boston, MA, An Introduction with Applications, (2001)
  • 20. Kac, V., Cheung, P.: Quantum Calculus. Universitext. Springer-Verlag, New York (2002)
  • 21. Abdeldaim, A., El-Deeb, A.A.: On generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro-differential...
  • 22. Agarwal, R.P., Bohner, M., Peterson, A.: Inequalities on time scales: a survey. Math. Inequal. Appl. 4(4), 535–557 (2001)
  • 23. El-Deeb, A.A.: On some generalizations of nonlinear dynamic inequalities on time scales and their applications. Appl. Anal. Discret. Math....
  • 24. El-Deeb, A. A., El-Sennary, H. A., Khan, Z. A.: Some Steffensen-type dynamic inequalities on time scales. Adv. Difference Equ., 246, (2019)
  • 25. Agarwal, R., O’Regan, D., Saker, S.: Dynamic inequalities on time scales. Springer, Cham (2014)
  • 26. Hilscher, R.: A time scales version of a Wirtinger-type inequality and applications. J. Comput. Appl. Math. 141(1–2), 219–226 (2002)
  • 27. Rˇ ehák, P.: Hardy inequality on time scales and its application to half-linear dynamic equations. J. Inequal. Appl. 5, 495–507 (2005)
  • 28. Rˇ ehák, P.: Hardy inequality on time scales and its application to half-linear dynamic equations. J. Inequal. Appl. 5, 495–507 (2005)
  • 29. Saker, S.H., O’Regan, D., Agarwal, R.: Generalized hardy, copson, leindler and bennett inequalities on time scales. Math. Nachr. 287(5–6),...
  • 30. Ozkan, U.M.,Yildirim, H.: Hardy-Knopp-type inequalities on time scales.Dynam. Syst.Appl. 17(3–4), 477–486 (2008)
  • 31. El-Deeb, A.A., Mohamed, K.A., Baleanu, D., Rezk, H.M.:Weighted dynamic hardy-type inequalities involving many functions on arbitrary time...
  • 32. El-Deeb, A.A., Makharesh, S.D., Almarri, B.: Some new inverse hilbert inequalities on time scales. Symmetry 14(11), 2234 (2022)
  • 33. El-Deeb, Ahmed A., Awrejcewicz, Jan: Diamond-α hardy-type inequalities on time scales. Symmetry 14(10), 2047 (2022)
  • 34. El-Deeb, A.A.: Novel dynamic hardy-type inequalities on time scales.Math.Method. Appl. Sci. 46(5), 5299–5313 (2023)
  • 35. Saker, S.H., Alzabut, J., Saied, A.I., O’Regan, D.: New characterizations of weights on dynamic inequalities involving a hardy operator....
  • 36. Saker, S.H., Saied, A.I., Krni´c, M.: Some new dynamic hardy-type inequalities with kernels involving monotone functions. Revista de la...
  • 37. Saker, S.H., Awwad, E., Saied, A.: Some new dynamic inequalities involving monotonic functions on time scales. J. Funct. Sp. 2019(1),...
  • 38. Rezk, H.M., Saied, A.I., AlNemer, G., Zakarya, M.: On Hardy-knopp type inequalities with kernels via time scale calculus. J. Math. 2022(1),...
  • 39. Awwad, E., Saied, A.I.: Some new multidimensional Hardy-type inequalities with general kernels on time scales. J. Math. Inequal. 16(1),...
  • 40. Saker, S.H., Saied, A.I., Anderson, D.R.:Somenewcharacterizations of weights in dynamic inequalities involving monotonic functions. Qual....
  • 41. Bohner, M.,Guseinov, GSh.: Multiple integration on time scales.Dynam. Syst.Appl. 14(3–4), 579–606 (2005)
  • 42. Agarwal, Ravi P.:DonalO’Regan, SamirHSaker, et al. Hardy type inequalities on time scales. Springer, (2016)
  • 43. Kaijser, Sten, Persson, Lars-Erik., Öberg, Anders: On carleman and knopp’s inequalities. J. Approx. Theory 117(1), 140–151 (2002)