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Year 2022, Volume: 51 Issue: 1, 48 - 73, 14.02.2022

Zeynep Kayar , Billur Kaymakçalan

https://doi.org/10.15672/hujms.928390
Cited By: 5

Abstract

References

  • [1] R. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl. 4 (4), 535-557, 2001.
  • [2] R.P. Agarwal, R.R. Mahmoud, S. Saker and C. Tunç, New generalizations of Németh- Mohapatra type inequalities on time scales, Acta Math. Hungar. 152 (2), 383-403, 2017.
  • [3] R. Agarwal, D. O’Regan and S. Saker, Dynamic Inequalities on Time Scales, Springer, Cham, 2014.
  • [4] R. Agarwal, D. O’Regan and S. Saker, Hardy Type Inequalities on Time Scales, Springer, Cham, 2016.
  • [5] M.R.S. Ammi, R.A.C. Ferreira, and D.F.M. Torres, Diamond-alpha Jensen’s inequality on time scales, J. Inequal. Appl. 2008 (Art. ID 576876), 1-13, 2008.
  • [6] D.R. Anderson, Time-scale integral inequalities, J. Inequal. Pure Appl. Math. 6 (3), Article 66, 1-15, 2005.
  • [7] N. Atasever, B. Kaymakçalan, G. Lešaja and K. Taş, Generalized diamond-alpha dynamic Opial inequalities, Adv. Difference Equ. 2012 (109), 1-9, 2012.
  • [8] F.M. Atici and G.S. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (1-2), 75-99, 2002.
  • [9] A.A. Balinsky, W.D. Evans and R.T. Lewis, The Analysis and Geometry of Hardy’s Inequality, Springer International Publishing, Switzerland, 2015.
  • [10] P.R. Beesack, Hardy’s inequality and its extensions, Pacific J. Math. 11 (1), 39-61, 1961.
  • [11] G. Bennett, Some elementary inequalities, Quart. J. Math. Oxford Ser. (2) 38 (152), 401-425, 1987.
  • [12] M. Bohner and O. Duman, Opial-type inequalities for diamond-alpha derivatives and integrals on time scales, Differ. Equ. Dyn. Syst. 18 (1-2), 229237, 2010.
  • [13] M. Bohner, R. Mahmoud and S.H. Saker, Discrete, continuous, delta, nabla, and diamond-alpha Opial inequalities, Math. Inequal. Appl. 18 (3), 923-940, 2015.
  • [14] M. Bohner, R.R. Mahmoud and S.H. Saker, Improvements of dynamic Opial-type inequalities and applications, Dynam. Syst. Appl. 24, 229-242, 2015.
  • [15] M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction With Applications, Birkhäuser Boston, Inc., Boston, MA, 2001.
  • [16] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Inc., Boston, MA, 2003.
  • [17] M.J. Bohner and S.H. Saker, Sneak-out principle on time scales, J. Math. Inequal. 10 (2), 393403, 2016.
  • [18] M.J. Bohner and S.H. Saker, Gehring Inequalities on Time Scales, J. Comput. Anal. Appl. 28 (1), 11-23, 2020.
  • [19] Y.-M. Chu, Q. Xu, and X.-M. Zhang, A note on Hardy’s inequality, J. Inequal. Appl. 2014 (271), 1-10, 2014.
  • [20] E.T. Copson, Note on series of positive terms, J. London Math. Soc. 3 (1), 49-51, 1928.
  • [21] E.T. Copson, Some integral inequalities, Proc. Roy. Soc. Edinburgh Sect. A 75 (2), 157-164, 1976.
  • [22] A.A. El-Deeb, H.A. Elsennary and Z.A. Khan, Some reverse inequalities of Hardy type on time scales, Adv. Difference Equ. 2020 (402), 1-18, 2020.
  • [23] A.A. El-Deeb, H.A. Elsennary and B. Dumitru, Some new Hardy-type inequalities on time scales, Adv. Difference Equ. 2020 (441), 1-22, 2020.
  • [24] P. Gao and H.Y. Zhao, On Copson’s inequalities for $0<p<1$, J. Inequal. Appl. 2020 (72), 1-13, 2020.
  • [25] G.S. Guseinov and B. Kaymakçalan, Basics of Riemann delta and nabla integration on time scales, J. Difference Equ. Appl. 8 (11), 1001-1017, 2002.
  • [26] M. Gürses, G.S. Guseinov and B. Silindir, Integrable equations on time scales, J. Math. Phys. 46 (11), 113510, 1-22, 2005.
  • [27] A.F. Güvenilir, B. Kaymakçalan and N.N. Pelen, Constantin’s inequality for nabla and diamond-alpha derivative, J. Inequal. Appl. 2015 (167), 1-17, 2015.
  • [28] G.H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (3-4), 314-317, 1920.
  • [29] G.H. Hardy, Notes on some points in the integral calculus, LX. An inequality between integrals, Messenger Math. 54 (3), 150-156, 1925.
  • [30] G.H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, London, 1934.
  • [31] M.M. Iddrisu, A.C. Okpoti and A.K. Gbolagade, Some proofs of the classical integral Hardy inequality, Korean J. Math. 22 (3), 407-417, 2014.
  • [32] S. Iqbal, M.J.S. Sahir and M. Samraiz, Symmetric Rogers-Hölder’s inequalities on diamond-alpha calculus, Int. J. Nonlinear Anal. Appl. 9 (2), 9-19, 2018.
  • [33] Z. Kayar and B. Kaymakçalan, Hardy-Copson type inequalities for nabla time scale calculus, Turk. J. Math. 45 (2), 1040-1064, 2021.
  • [34] Z. Kayar and B. Kaymakçalan, Some extended nabla and delta HardyCopson type inequalities with applications in oscillation theory, Bull. Iran. Math. Soc., accepted, doi:10.1007/s41980-021-00651-2.
  • [35] Z. Kayar and B. Kaymakçalan, Complements of nabla and delta Hardy-Copson type inequalities and their applications, submitted.
  • [36] Z. Kayar and B. Kaymakçalan, Extensions of diamond-alpha Hardy-Copson type dynamic inequalities and their applications to oscillation theory, Dyn. Syst. Appl. 30 (7), 1180-1209, 2021.
  • [37] Z. Kayar and B. Kaymakçalan, Applications of the novel diamond-alpha Hardy- Copson type dynamic inequalities to half linear difference equations, J. Differ. Equ. Appl., accepted.
  • [38] Z. Kayar and B. Kaymakçalan, Novel diamond-alpha Bennett-Leindler type dynamic inequalities, submitted.
  • [39] Z. Kayar, B. Kaymakçalan and N.N. Pelen, Diamond-alpha Bennett-Leindler type dynamic inequalities, Math. Methods Appl. Sci., accepted.
  • [40] Z. Kayar, B. Kaymakçalan and N.N. Pelen, Bennett-Leindler type inequalities for time scale nabla calculus, Mediterr. J. Math. 18 (14), (2021).
  • [41] Z. Kayar and B. Kaymakçalan, The complementary nabla Bennett-Leindler type inequalities, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., accepted.
  • [42] A. Kufner, L. Maligranda and L.E. Persson, The Hardy Inequality. About Its History and Some Related Results, Vydavatelský Servis, Pilsen, 2007.
  • [43] A. Kufner, L.E. Persson and N. Samko, Weighted Inequalities of Hardy Type, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
  • [44] P. Lefèvre, A short direct proof of the discrete Hardy inequality, Arch. Math. (Basel). 114 (2), 195-198, 2020.
  • [45] L. Leindler, Some inequalities pertaining to Bennett’s results, Acta Sci. Math. (Szeged). 58 (1-4), 261-279, 1993.
  • [46] L. Leindler, A Theorem of Hardy-Bennett-Type, Acta Math. Hungar. 78 (4), 315325, 1998.
  • [47] Z.-W. Liao, Discrete Hardy-type inequalities, Adv. Nonlinear Stud. 15 (4), 805-834, 2015.
  • [48] A.B. Malinowska and D.F.M. Torres, On the diamond-alpha Riemann integral and mean value theorems on time scales, Dynam. Syst. Appl. 18 (3-4), 469-481, 2009.
  • [49] N. Masmoudi, About the Hardy Inequality, in: An Invitation to Mathematics. From Competitions to Research, Springer, Heidelberg, 2011.
  • [50] T.Z. Mirković, Dynamic Opial diamond-alpha integral inequalities involving the power of a function, J. Inequal. Appl. 2017 (139), 1-10, 2017.
  • [51] D. Mozyrska and D.F.M. Torres, A study of diamond-alpha dynamic equations on regular time scales, Afr. Diaspora J. Math. (N.S.) 8 (1), 35-47, 2009.
  • [52] E.N. Nikolidakis, A sharp integral Hardy type inequality and applications to Muckenhoupt weights on $\mathbb{R}$, Ann. Acad. Sci. Fenn. Math. 39 (2), 887-896, 2014.
  • [53] U.M. Özkan, M.Z. Sarikaya and H. Yildirim, Extensions of certain integral inequalities on time scales, Appl. Math. Lett. 21 (10), 993-1000, 2008.
  • [54] B.G. Pachpatte, On Some Generalizations of Hardys Integral Inequality, J. Math. Anal. Appl. 234 (1), 15-30, 1999.
  • [55] J. Pečarić and Ž. Hanjš, On some generalizations of inequalities given by B. G. Pachpatte, An. Şttiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 45 (1), 103-114, 1999.
  • [56] N.N. Pelen, Hardy-Sobolev-Mazya inequality for nabla time scale calculus, Eskişehir Technical University Journal of Science and Technology B - Theoretical Sciences 7 (2), 133-145, 2019.
  • [57] J.W. Rogers Jr. and Q. Sheng, Notes on the diamond-alpha dynamic derivative on time scales, J. Math. Anal. Appl. 326 (1), 228-241, 2007.
  • [58] S.H. Saker, Dynamic inequalities on time scales: A survey, J. Fractional Calc. & Appl. 3(S) (2), 1-36, 2012.
  • [59] S.H. Saker and R.R. Mahmoud, A connection between weighted Hardy’s inequality and half-linear dynamic equations, Adv. Difference Equ. 2014 (129), 1-15, 2019.
  • [60] S.H. Saker, R.R. Mahmoud and A. Peterson, Some Bennett-Copson type inequalities on time scales, J. Math. Inequal. 10 (2), 471-489, 2016.
  • [61] S.H. Saker, R.R. Mahmoud, M.M. Osman and R.P. Agarwal, Some new generalized forms of Hardy’s type inequality on time scales, Math. Inequal. Appl. 20 (2), 459-481, 2017.
  • [62] S.H. Saker, D. O’Regan and R.P. Agarwal, Dynamic inequalities of Hardy and Copson type on time scales, Analysis 34 (4), 391-402, 2014.
  • [63] S.H. Saker, D. O’Regan and R.P. Agarwal, Generalized Hardy, Copson, Leindler and Bennett inequalities on time scales, Math. Nachr. 287 (5-6), 686-698, 2014.
  • [64] S.H. Saker, M.M. Osman, D. O’Regan and R.P. Agarwal, Inequalities of Hardy type and generalizations on time scales, Analysis 38 (1), 4762, 2018.
  • [65] S.H. Saker, R.R. Mahmoud and A. Peterson, A unified approach to Copson and Beesack type inequalities on time scales, Math. Inequal. Appl. 21 (4), 985-1002, 2018.
  • [66] Q. Sheng, M. Fadag, J. Henderson and J.M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl. 7 (3), 395-413, 2006.

Diamond alpha Hardy-Copson type dynamic inequalities

Year 2022, Volume: 51 Issue: 1, 48 - 73, 14.02.2022

Zeynep Kayar , Billur Kaymakçalan

https://doi.org/10.15672/hujms.928390
Cited By: 5

Abstract

In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.

Keywords

Diamond alpha calculus Hardy’s inequality Copson’s inequality

References

  • [1] R. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl. 4 (4), 535-557, 2001.
  • [2] R.P. Agarwal, R.R. Mahmoud, S. Saker and C. Tunç, New generalizations of Németh- Mohapatra type inequalities on time scales, Acta Math. Hungar. 152 (2), 383-403, 2017.
  • [3] R. Agarwal, D. O’Regan and S. Saker, Dynamic Inequalities on Time Scales, Springer, Cham, 2014.
  • [4] R. Agarwal, D. O’Regan and S. Saker, Hardy Type Inequalities on Time Scales, Springer, Cham, 2016.
  • [5] M.R.S. Ammi, R.A.C. Ferreira, and D.F.M. Torres, Diamond-alpha Jensen’s inequality on time scales, J. Inequal. Appl. 2008 (Art. ID 576876), 1-13, 2008.
  • [6] D.R. Anderson, Time-scale integral inequalities, J. Inequal. Pure Appl. Math. 6 (3), Article 66, 1-15, 2005.
  • [7] N. Atasever, B. Kaymakçalan, G. Lešaja and K. Taş, Generalized diamond-alpha dynamic Opial inequalities, Adv. Difference Equ. 2012 (109), 1-9, 2012.
  • [8] F.M. Atici and G.S. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (1-2), 75-99, 2002.
  • [9] A.A. Balinsky, W.D. Evans and R.T. Lewis, The Analysis and Geometry of Hardy’s Inequality, Springer International Publishing, Switzerland, 2015.
  • [10] P.R. Beesack, Hardy’s inequality and its extensions, Pacific J. Math. 11 (1), 39-61, 1961.
  • [11] G. Bennett, Some elementary inequalities, Quart. J. Math. Oxford Ser. (2) 38 (152), 401-425, 1987.
  • [12] M. Bohner and O. Duman, Opial-type inequalities for diamond-alpha derivatives and integrals on time scales, Differ. Equ. Dyn. Syst. 18 (1-2), 229237, 2010.
  • [13] M. Bohner, R. Mahmoud and S.H. Saker, Discrete, continuous, delta, nabla, and diamond-alpha Opial inequalities, Math. Inequal. Appl. 18 (3), 923-940, 2015.
  • [14] M. Bohner, R.R. Mahmoud and S.H. Saker, Improvements of dynamic Opial-type inequalities and applications, Dynam. Syst. Appl. 24, 229-242, 2015.
  • [15] M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction With Applications, Birkhäuser Boston, Inc., Boston, MA, 2001.
  • [16] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Inc., Boston, MA, 2003.
  • [17] M.J. Bohner and S.H. Saker, Sneak-out principle on time scales, J. Math. Inequal. 10 (2), 393403, 2016.
  • [18] M.J. Bohner and S.H. Saker, Gehring Inequalities on Time Scales, J. Comput. Anal. Appl. 28 (1), 11-23, 2020.
  • [19] Y.-M. Chu, Q. Xu, and X.-M. Zhang, A note on Hardy’s inequality, J. Inequal. Appl. 2014 (271), 1-10, 2014.
  • [20] E.T. Copson, Note on series of positive terms, J. London Math. Soc. 3 (1), 49-51, 1928.
  • [21] E.T. Copson, Some integral inequalities, Proc. Roy. Soc. Edinburgh Sect. A 75 (2), 157-164, 1976.
  • [22] A.A. El-Deeb, H.A. Elsennary and Z.A. Khan, Some reverse inequalities of Hardy type on time scales, Adv. Difference Equ. 2020 (402), 1-18, 2020.
  • [23] A.A. El-Deeb, H.A. Elsennary and B. Dumitru, Some new Hardy-type inequalities on time scales, Adv. Difference Equ. 2020 (441), 1-22, 2020.
  • [24] P. Gao and H.Y. Zhao, On Copson’s inequalities for $0<p<1$, J. Inequal. Appl. 2020 (72), 1-13, 2020.
  • [25] G.S. Guseinov and B. Kaymakçalan, Basics of Riemann delta and nabla integration on time scales, J. Difference Equ. Appl. 8 (11), 1001-1017, 2002.
  • [26] M. Gürses, G.S. Guseinov and B. Silindir, Integrable equations on time scales, J. Math. Phys. 46 (11), 113510, 1-22, 2005.
  • [27] A.F. Güvenilir, B. Kaymakçalan and N.N. Pelen, Constantin’s inequality for nabla and diamond-alpha derivative, J. Inequal. Appl. 2015 (167), 1-17, 2015.
  • [28] G.H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (3-4), 314-317, 1920.
  • [29] G.H. Hardy, Notes on some points in the integral calculus, LX. An inequality between integrals, Messenger Math. 54 (3), 150-156, 1925.
  • [30] G.H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, London, 1934.
  • [31] M.M. Iddrisu, A.C. Okpoti and A.K. Gbolagade, Some proofs of the classical integral Hardy inequality, Korean J. Math. 22 (3), 407-417, 2014.
  • [32] S. Iqbal, M.J.S. Sahir and M. Samraiz, Symmetric Rogers-Hölder’s inequalities on diamond-alpha calculus, Int. J. Nonlinear Anal. Appl. 9 (2), 9-19, 2018.
  • [33] Z. Kayar and B. Kaymakçalan, Hardy-Copson type inequalities for nabla time scale calculus, Turk. J. Math. 45 (2), 1040-1064, 2021.
  • [34] Z. Kayar and B. Kaymakçalan, Some extended nabla and delta HardyCopson type inequalities with applications in oscillation theory, Bull. Iran. Math. Soc., accepted, doi:10.1007/s41980-021-00651-2.
  • [35] Z. Kayar and B. Kaymakçalan, Complements of nabla and delta Hardy-Copson type inequalities and their applications, submitted.
  • [36] Z. Kayar and B. Kaymakçalan, Extensions of diamond-alpha Hardy-Copson type dynamic inequalities and their applications to oscillation theory, Dyn. Syst. Appl. 30 (7), 1180-1209, 2021.
  • [37] Z. Kayar and B. Kaymakçalan, Applications of the novel diamond-alpha Hardy- Copson type dynamic inequalities to half linear difference equations, J. Differ. Equ. Appl., accepted.
  • [38] Z. Kayar and B. Kaymakçalan, Novel diamond-alpha Bennett-Leindler type dynamic inequalities, submitted.
  • [39] Z. Kayar, B. Kaymakçalan and N.N. Pelen, Diamond-alpha Bennett-Leindler type dynamic inequalities, Math. Methods Appl. Sci., accepted.
  • [40] Z. Kayar, B. Kaymakçalan and N.N. Pelen, Bennett-Leindler type inequalities for time scale nabla calculus, Mediterr. J. Math. 18 (14), (2021).
  • [41] Z. Kayar and B. Kaymakçalan, The complementary nabla Bennett-Leindler type inequalities, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., accepted.
  • [42] A. Kufner, L. Maligranda and L.E. Persson, The Hardy Inequality. About Its History and Some Related Results, Vydavatelský Servis, Pilsen, 2007.
  • [43] A. Kufner, L.E. Persson and N. Samko, Weighted Inequalities of Hardy Type, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
  • [44] P. Lefèvre, A short direct proof of the discrete Hardy inequality, Arch. Math. (Basel). 114 (2), 195-198, 2020.
  • [45] L. Leindler, Some inequalities pertaining to Bennett’s results, Acta Sci. Math. (Szeged). 58 (1-4), 261-279, 1993.
  • [46] L. Leindler, A Theorem of Hardy-Bennett-Type, Acta Math. Hungar. 78 (4), 315325, 1998.
  • [47] Z.-W. Liao, Discrete Hardy-type inequalities, Adv. Nonlinear Stud. 15 (4), 805-834, 2015.
  • [48] A.B. Malinowska and D.F.M. Torres, On the diamond-alpha Riemann integral and mean value theorems on time scales, Dynam. Syst. Appl. 18 (3-4), 469-481, 2009.
  • [49] N. Masmoudi, About the Hardy Inequality, in: An Invitation to Mathematics. From Competitions to Research, Springer, Heidelberg, 2011.
  • [50] T.Z. Mirković, Dynamic Opial diamond-alpha integral inequalities involving the power of a function, J. Inequal. Appl. 2017 (139), 1-10, 2017.
  • [51] D. Mozyrska and D.F.M. Torres, A study of diamond-alpha dynamic equations on regular time scales, Afr. Diaspora J. Math. (N.S.) 8 (1), 35-47, 2009.
  • [52] E.N. Nikolidakis, A sharp integral Hardy type inequality and applications to Muckenhoupt weights on $\mathbb{R}$, Ann. Acad. Sci. Fenn. Math. 39 (2), 887-896, 2014.
  • [53] U.M. Özkan, M.Z. Sarikaya and H. Yildirim, Extensions of certain integral inequalities on time scales, Appl. Math. Lett. 21 (10), 993-1000, 2008.
  • [54] B.G. Pachpatte, On Some Generalizations of Hardys Integral Inequality, J. Math. Anal. Appl. 234 (1), 15-30, 1999.
  • [55] J. Pečarić and Ž. Hanjš, On some generalizations of inequalities given by B. G. Pachpatte, An. Şttiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 45 (1), 103-114, 1999.
  • [56] N.N. Pelen, Hardy-Sobolev-Mazya inequality for nabla time scale calculus, Eskişehir Technical University Journal of Science and Technology B - Theoretical Sciences 7 (2), 133-145, 2019.
  • [57] J.W. Rogers Jr. and Q. Sheng, Notes on the diamond-alpha dynamic derivative on time scales, J. Math. Anal. Appl. 326 (1), 228-241, 2007.
  • [58] S.H. Saker, Dynamic inequalities on time scales: A survey, J. Fractional Calc. & Appl. 3(S) (2), 1-36, 2012.
  • [59] S.H. Saker and R.R. Mahmoud, A connection between weighted Hardy’s inequality and half-linear dynamic equations, Adv. Difference Equ. 2014 (129), 1-15, 2019.
  • [60] S.H. Saker, R.R. Mahmoud and A. Peterson, Some Bennett-Copson type inequalities on time scales, J. Math. Inequal. 10 (2), 471-489, 2016.
  • [61] S.H. Saker, R.R. Mahmoud, M.M. Osman and R.P. Agarwal, Some new generalized forms of Hardy’s type inequality on time scales, Math. Inequal. Appl. 20 (2), 459-481, 2017.
  • [62] S.H. Saker, D. O’Regan and R.P. Agarwal, Dynamic inequalities of Hardy and Copson type on time scales, Analysis 34 (4), 391-402, 2014.
  • [63] S.H. Saker, D. O’Regan and R.P. Agarwal, Generalized Hardy, Copson, Leindler and Bennett inequalities on time scales, Math. Nachr. 287 (5-6), 686-698, 2014.
  • [64] S.H. Saker, M.M. Osman, D. O’Regan and R.P. Agarwal, Inequalities of Hardy type and generalizations on time scales, Analysis 38 (1), 4762, 2018.
  • [65] S.H. Saker, R.R. Mahmoud and A. Peterson, A unified approach to Copson and Beesack type inequalities on time scales, Math. Inequal. Appl. 21 (4), 985-1002, 2018.
  • [66] Q. Sheng, M. Fadag, J. Henderson and J.M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl. 7 (3), 395-413, 2006.
There are 66 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Zeynep Kayar 0000-0002-8309-7930

Billur Kaymakçalan This is me 0000-0002-0763-9128

Publication Date February 14, 2022
Published in Issue Year 2022 Volume: 51 Issue: 1

Cite

APA Kayar, Z., & Kaymakçalan, B. (2022). Diamond alpha Hardy-Copson type dynamic inequalities. Hacettepe Journal of Mathematics and Statistics, 51(1), 48-73. https://doi.org/10.15672/hujms.928390
AMA Kayar Z, Kaymakçalan B. Diamond alpha Hardy-Copson type dynamic inequalities. Hacettepe Journal of Mathematics and Statistics. February 2022;51(1):48-73. doi:10.15672/hujms.928390
Chicago Kayar, Zeynep, and Billur Kaymakçalan. “Diamond Alpha Hardy-Copson Type Dynamic Inequalities”. Hacettepe Journal of Mathematics and Statistics 51, no. 1 (February 2022): 48-73. https://doi.org/10.15672/hujms.928390.
EndNote Kayar Z, Kaymakçalan B (February 1, 2022) Diamond alpha Hardy-Copson type dynamic inequalities. Hacettepe Journal of Mathematics and Statistics 51 1 48–73.
IEEE Z. Kayar and B. Kaymakçalan, “Diamond alpha Hardy-Copson type dynamic inequalities”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, pp. 48–73, 2022, doi: 10.15672/hujms.928390.
ISNAD Kayar, Zeynep - Kaymakçalan, Billur. “Diamond Alpha Hardy-Copson Type Dynamic Inequalities”. Hacettepe Journal of Mathematics and Statistics 51/1 (February 2022), 48-73. https://doi.org/10.15672/hujms.928390.
JAMA Kayar Z, Kaymakçalan B. Diamond alpha Hardy-Copson type dynamic inequalities. Hacettepe Journal of Mathematics and Statistics. 2022;51:48–73.
MLA Kayar, Zeynep and Billur Kaymakçalan. “Diamond Alpha Hardy-Copson Type Dynamic Inequalities”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, 2022, pp. 48-73, doi:10.15672/hujms.928390.
Vancouver Kayar Z, Kaymakçalan B. Diamond alpha Hardy-Copson type dynamic inequalities. Hacettepe Journal of Mathematics and Statistics. 2022;51(1):48-73.

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