SciELO - Scientific Electronic Library Online

vol.39 issue1A cryptography method based on hyperbolicbalancing and Lucas-balancing functionsThe total double geodetic number of a graph author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand




Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.39 no.1 Antofagasta Feb. 2020 


Some refinements to Hölder’s inequality and applications

Mohamed Akkouchi1 

Mohamed Amine Ighachane2

1 Cadi Ayyad University, Dept. of Mathematics, Marrakech, Morocco. E-mail:

2 Cadi Ayyad University, Dept. of Mathematics, Marrakech, Morocco. E-mail:


We establish some new refinements to the Hölder inequality. We then apply them to provide some refinements to the extended Euler’s gamma and beta functions. As another application of our results, we give a new proof of the equivalence between the Hölder inequality and the Cauchy-Schwarz inequality.

Keywords: Inequalities; Young’s inequality; Cauchy-Schwarz inequality; Inequalities for extended Beta and Gamma functions

Texto completo disponible sólo en PDF.

Full text available only in PDF format.


The authors would like to express their deep thanks to the anonymous referee for his/her comments and suggestions on the initial version of the manuscript which lead to the improvement of this paper.


[1] M. Akkouchi, “Inequalities for real random variables connected with Jensen’s inequality and applications”, Rivista di matematica della Universitá di Parma, vol. 6, no. 3, pp. 113-125, 2000. [On line]. Available: ]

[2] M. Akkouchi, “Cauchy-Schwarz inequality implies Hölder’s inequality”, RGMIA Research report collection, vol. 21, Art. ID. 48, 2018. [On line]. Available: ]

[3] J. M. Aldaz, “Self improvement of the inequality between arithmetic and geometric means”, Journal of mathematical inequalities, vol. 3, no. 2, pp. 213-216, 2009, doi: 10.7153/jmi-03-21. [ Links ]

[4] R. M. Ali, S. R. Mondal, and K. S. Nisar, “Monotonicity properties of the generalized Struve functions”, Journal of korean mathematical society, vol. 54, no. 2, pp. 575-598, Mar. 2017, doi: 10.4134/JKMS.j160137. [ Links ]

[5] P. K. Bhandari and S. K. Bissu, “Inequalities via Hölder’s inequality”, Scholars journal of research in mathematics and computer science, vol. 2, no. 2, pp. 124-129, 2018. [On line]. Available: ]

[6] M. A. Chaudhry, A. Qadir, M. Rafique, and S. M. Zubair, “Extension of Euler’s beta function”, Journal of computational and applied mathematics, vol. 78, no. 1, pp. 19-32, Feb. 1997, doi: 10.1016/S0377-0427(96)00102-1. [ Links ]

[7] M. A. Chaudhry and S. M. Zubair, On a class of incomplete gamma functions with applications. Boca Raton, FL: Chapman & Hall/CRC, 2002. [ Links ]

[8] S. S. Dragomir, R. P. Agarwal and N. S. Barnett, “Inequalities for beta and gamma functions via some classical and new integral inequalities”, Journal of inequalities and applications, vol. 5, no. 2, pp. 103-165, 2000, doi: 10.1155/S1025583400000084. [ Links ]

[9] C. Finol and M. Wojtowicz, “Cauchy-Schwarz and Hölder’s inequalities are equivalent”, Divulgaciones matemáticas, vol. 15, no. 2, pp. 143-147, 2007. [On line]. Available: ]

[10] C. A. Infantozzi, “An introduction to relations among inequalities. The November Meeting in Cleveland, Ohio November 25, 1972”, Notices of the American mathematical society, no. 141, pp. A819-A820, 1972. [On line]. Available: ]

[11] Y-C. Li and S-Y. Shaw, “A proof of Hölder’s inequality using the Cauchy-Schwarz inequality”, Journal of inequalities in pure and applied mathematics, vol. 7, no. 2, Art. ID. 62, 2006. [On line]. Available: ]

[12] A. W. Marshall and I. Olkin, Inequalities: theory of majorization and its applications. New York, NY: Academic Press, 1979. [ Links ]

[13] D. S. Mtirinovic, J. E. Pečarić and A. M. Fink, Classical and new inequalities in analysis. Dordrecht: Kluwer Academic Publishers, 1993. [ Links ]

[14] K. S. Nisar, S. R. Mondal, and J. Choi, “Certain inequalities involving the k-Struve function”, Journal of inequalities and applications , Art. ID. 71, Apr. 2017, doi: 10.1186/s13660-017-1343-x. [ Links ]

[15] K. S. Nisar , F. Qi, G. Rahman, S. Mubeen, and M. Arshad, “Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function”, Journal of inequalities and applications , Art. ID. 135, Jun. 2018, doi: 10.1186/s13660-018-1717-8. [ Links ]

Received: January 2019; Accepted: May 2019

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License