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Year 2021, Volume: 5 Issue: 1, 1 - 6, 31.03.2021
https://doi.org/10.31197/atnaa.817804

Abstract

References

  • [1] D. R. Anderson, Young’s integral inequality on time scales revisited, J. Inequal. Pure Appl. Math. 8 (2007), no. 3, Art. 64; http://www.emis.de/journals/JIPAM/ article876.html.
  • [2] R. P. Boas Jr. and M. B. Marcus, Generalizations of Young’s inequality, J. Math. Anal. Appl. 46 (1974), no. 1, 36–40; https://doi.org/10.1016/0022-247X(74)90279-0.
  • [3] R. P. Boas Jr. and M. B. Marcus, Inequalities involving a function and its inverse, SIAM J.Math. Anal. 4 (1973), 585–591; https://doi.org/10.1137/0504051.
  • [4] R. Cooper, Notes on certain inequalities: (1); Generalization of an inequality of W. H.Young, J. London Math. Soc. 2 (1927), no. 1, 17–21;https://doi.org/10.1112/jlms/s1-2.1.17.
  • [5] R. Cooper, Notes on certain inequalities: II, J. London Math. Soc. 2 (1927), no. 3, 159–163; https://doi.org/10.1112/jlms/s1-2.3.159.
  • [6] F. Cunningham, Jr. and N. Grossman, On Young’s inequality, Amer. Math. Monthly 78 (1971), no. 7, 781–783; https://doi.org/10.2307/2318018.
  • [7] J. B. Diaz and F. T. Metcalf, An analytic proof of Young’s inequality, Amer. Math. Monthly 77 (1970), no. 6, 603–609;https://doi.org/10.2307/2316736.
  • [8] A. Hoorfar and F. Qi, A new refinement of Young’s inequality, Math. Inequal. Appl. 11 (2008), no. 4, 689–692; https://doi.org/10.7153/mia-11-58.
  • [9] I. C. Hsu, On a converse of Young’s inequality, Proc. Amer. Math. Soc. 33 (1972), 107–108; https://doi.org/10.2307/2038179.
  • [10] J. Jaksetic and J. Pecaric, An estimation of Young inequality, Asian-Eur. J. Math. 2 (2009), no. 4, 593–604; https://doi.org/10.1142/S1793557109000509.
  • [11] J. Jaksetic and J. Pecaric, A note on Young inequality, Math. Inequal. Appl. 13 (2010), no. 1, 43–48; https://doi.org/10.7153/mia-13-03.
  • [12] I. A. Lackovic, A note on a converse of Young’s inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 461497 (1974), 73–76.
  • [13] D. S. Mitrinovic, Analytic Inequalities, In cooperation with P. M. Vasi´ c, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970.
  • [14] D. S. Mitrinovic, J. E. Pecari´ c, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993; http://dx.doi.org/10.1007/978-94-017-1043-5.
  • [15] F.-C. Mitroi and C. P. Niculescu, An extension of Young’s inequality, Abstr. Appl. Anal. 2011, Art. ID 162049, 18 pages; https://doi.org/10.1155/2011/162049.
  • [16] A. Oppenheim, Note on Mr. Cooper’s generalization of Young’s inequality, J. London Math. Soc. 2 (1927), no. 1, 21–23; https://doi.org/10.1112/jlms/s1-2.1.21.
  • [17] Z. Pales, A general version of Young’s inequality, Arch. Math. (Basel) 58 (1992), no. 4,360–365; https://doi.org/10.1007/BF01189925.
  • [18] Z. Pales, A generalization of Young’s inequality, in General Inequalities, 5 (Oberwolfach, 1986), 471–472, Internat. Schriftenreihe Numer. Math., 80, Birkhäuser, Basel, 1987.
  • [19] Z. Pales, On Young-type inequalities, Acta Sci. Math. (Szeged) 54 (1990), no. 3-4, 327–338.
  • [20] F. D. Parker, Integrals of inverse functions, Amer. Math. Monthly 62 (1955), no. 6, 439–440; https://doi.org/10.2307/2307006.
  • [21] F. Qi, W.-H. Li, G.-S. Wu, and B.-N. Guo, Re?nements of Young's integral inequality via fundamental inequalities and mean value theorems for derivatives, Chapter 8 in: Hemen Dutta (ed.), Topics in Contemporary Mathematical Analysis and Applications, pp. 193-227, CRC Press, 2021; available online at https://doi.org/10.1201/9781003081197-8.
  • [22] D. Ruthing, On Young’s inequality, Internat. J. Math. Ed. Sci. Techn. 25 (1994), no. 2, 161–164; https://doi.org/10.1080/0020739940250201.
  • [23] T. Takahashi, Remarks on some inequalities, Tôhoku Math. J. 36 (1932), 99–106.
  • [24] J.-Q. Wang, B.-N. Guo, and F. Qi, Generalizations and applications of Young’s integral inequality by higher order derivatives, J. Inequal. Appl. 2019, Paper No. 243, 18 pages; https://doi.org/10.1186/s13660-019-2196-2.
  • [25] A. Witkowski, On Young inequality, J. Inequal. Pure Appl. Math. 7 (2006), no. 5, Art. 164; http://www.emis.de/journals/JIPAM/article782.html.
  • [26] F.-H. Wong, C.-C. Yeh, S.-L. Yu, and C.-H. Hong, Young’s inequality and related results on time scales, Appl. Math. Lett. 18 (2005), no. 9, 983–988; https://doi.org/10.1016/j.aml.2004.06.028.
  • [27] W. H. Young, On classes of summable functions and their Fourier series, Proc. Roy. Soc. London Ser. A 87 (1912), 225–229; https://doi.org/10.1098/rspa.1912.0076.
  • [28] L. Zhu, On Young’s inequality, Internat. J. Math. Ed. Sci. Tech. 35 (2004), no. 4, 601–603; https://doi.org/10.1080/00207390410001686698.

Geometric interpretations and reversed versions of Young's integral inequality

Year 2021, Volume: 5 Issue: 1, 1 - 6, 31.03.2021
https://doi.org/10.31197/atnaa.817804

Abstract

The authors retrospect Young's integral inequality and its geometric interpretation, recall a reversed version of Young's integral inequality, present a geometric interpretation of the reversed version of Young's integral inequality, and conclude a new reversed version of Young's integral inequality.

The authors retrospect Young's integral inequality and its geometric interpretation, recall a reversed version of Young's integral inequality, present a geometric interpretation of the reversed version of Young's integral inequality, and conclude a new reversed version of Young's integral inequality.                                                                                                                                                                                                                .

References

  • [1] D. R. Anderson, Young’s integral inequality on time scales revisited, J. Inequal. Pure Appl. Math. 8 (2007), no. 3, Art. 64; http://www.emis.de/journals/JIPAM/ article876.html.
  • [2] R. P. Boas Jr. and M. B. Marcus, Generalizations of Young’s inequality, J. Math. Anal. Appl. 46 (1974), no. 1, 36–40; https://doi.org/10.1016/0022-247X(74)90279-0.
  • [3] R. P. Boas Jr. and M. B. Marcus, Inequalities involving a function and its inverse, SIAM J.Math. Anal. 4 (1973), 585–591; https://doi.org/10.1137/0504051.
  • [4] R. Cooper, Notes on certain inequalities: (1); Generalization of an inequality of W. H.Young, J. London Math. Soc. 2 (1927), no. 1, 17–21;https://doi.org/10.1112/jlms/s1-2.1.17.
  • [5] R. Cooper, Notes on certain inequalities: II, J. London Math. Soc. 2 (1927), no. 3, 159–163; https://doi.org/10.1112/jlms/s1-2.3.159.
  • [6] F. Cunningham, Jr. and N. Grossman, On Young’s inequality, Amer. Math. Monthly 78 (1971), no. 7, 781–783; https://doi.org/10.2307/2318018.
  • [7] J. B. Diaz and F. T. Metcalf, An analytic proof of Young’s inequality, Amer. Math. Monthly 77 (1970), no. 6, 603–609;https://doi.org/10.2307/2316736.
  • [8] A. Hoorfar and F. Qi, A new refinement of Young’s inequality, Math. Inequal. Appl. 11 (2008), no. 4, 689–692; https://doi.org/10.7153/mia-11-58.
  • [9] I. C. Hsu, On a converse of Young’s inequality, Proc. Amer. Math. Soc. 33 (1972), 107–108; https://doi.org/10.2307/2038179.
  • [10] J. Jaksetic and J. Pecaric, An estimation of Young inequality, Asian-Eur. J. Math. 2 (2009), no. 4, 593–604; https://doi.org/10.1142/S1793557109000509.
  • [11] J. Jaksetic and J. Pecaric, A note on Young inequality, Math. Inequal. Appl. 13 (2010), no. 1, 43–48; https://doi.org/10.7153/mia-13-03.
  • [12] I. A. Lackovic, A note on a converse of Young’s inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 461497 (1974), 73–76.
  • [13] D. S. Mitrinovic, Analytic Inequalities, In cooperation with P. M. Vasi´ c, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970.
  • [14] D. S. Mitrinovic, J. E. Pecari´ c, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993; http://dx.doi.org/10.1007/978-94-017-1043-5.
  • [15] F.-C. Mitroi and C. P. Niculescu, An extension of Young’s inequality, Abstr. Appl. Anal. 2011, Art. ID 162049, 18 pages; https://doi.org/10.1155/2011/162049.
  • [16] A. Oppenheim, Note on Mr. Cooper’s generalization of Young’s inequality, J. London Math. Soc. 2 (1927), no. 1, 21–23; https://doi.org/10.1112/jlms/s1-2.1.21.
  • [17] Z. Pales, A general version of Young’s inequality, Arch. Math. (Basel) 58 (1992), no. 4,360–365; https://doi.org/10.1007/BF01189925.
  • [18] Z. Pales, A generalization of Young’s inequality, in General Inequalities, 5 (Oberwolfach, 1986), 471–472, Internat. Schriftenreihe Numer. Math., 80, Birkhäuser, Basel, 1987.
  • [19] Z. Pales, On Young-type inequalities, Acta Sci. Math. (Szeged) 54 (1990), no. 3-4, 327–338.
  • [20] F. D. Parker, Integrals of inverse functions, Amer. Math. Monthly 62 (1955), no. 6, 439–440; https://doi.org/10.2307/2307006.
  • [21] F. Qi, W.-H. Li, G.-S. Wu, and B.-N. Guo, Re?nements of Young's integral inequality via fundamental inequalities and mean value theorems for derivatives, Chapter 8 in: Hemen Dutta (ed.), Topics in Contemporary Mathematical Analysis and Applications, pp. 193-227, CRC Press, 2021; available online at https://doi.org/10.1201/9781003081197-8.
  • [22] D. Ruthing, On Young’s inequality, Internat. J. Math. Ed. Sci. Techn. 25 (1994), no. 2, 161–164; https://doi.org/10.1080/0020739940250201.
  • [23] T. Takahashi, Remarks on some inequalities, Tôhoku Math. J. 36 (1932), 99–106.
  • [24] J.-Q. Wang, B.-N. Guo, and F. Qi, Generalizations and applications of Young’s integral inequality by higher order derivatives, J. Inequal. Appl. 2019, Paper No. 243, 18 pages; https://doi.org/10.1186/s13660-019-2196-2.
  • [25] A. Witkowski, On Young inequality, J. Inequal. Pure Appl. Math. 7 (2006), no. 5, Art. 164; http://www.emis.de/journals/JIPAM/article782.html.
  • [26] F.-H. Wong, C.-C. Yeh, S.-L. Yu, and C.-H. Hong, Young’s inequality and related results on time scales, Appl. Math. Lett. 18 (2005), no. 9, 983–988; https://doi.org/10.1016/j.aml.2004.06.028.
  • [27] W. H. Young, On classes of summable functions and their Fourier series, Proc. Roy. Soc. London Ser. A 87 (1912), 225–229; https://doi.org/10.1098/rspa.1912.0076.
  • [28] L. Zhu, On Young’s inequality, Internat. J. Math. Ed. Sci. Tech. 35 (2004), no. 4, 601–603; https://doi.org/10.1080/00207390410001686698.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Feng Qi

Aying Wan This is me

Publication Date March 31, 2021
Published in Issue Year 2021 Volume: 5 Issue: 1

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