Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 4 (2019), 671-686.

Log-concavity of Hölder means and an application to geometric inequalities

Aurel I. Stan and Sergio D. Zapeta-Tzul

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Abstract

The log-concavity of the Hölder mean of two numbers, as a function of its index, is presented first. The notion of α-cevian of a triangle is introduced next, for any real number α. We use this property of the Hölder mean to find the smallest index p(α) such that the length of an α-cevian of a triangle is less than or equal to the p(α)-Hölder mean of the lengths of the two sides of the triangle that are adjacent to that cevian.

Article information

Source
Involve, Volume 12, Number 4 (2019), 671-686.

Dates
Received: 23 May 2018
Revised: 9 November 2018
Accepted: 15 November 2018
First available in Project Euclid: 30 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.involve/1559181659

Digital Object Identifier
doi:10.2140/involve.2019.12.671

Mathematical Reviews number (MathSciNet)
MR3941605

Zentralblatt MATH identifier
07072546

Subjects
Primary: 26A06: One-variable calculus 26D99: None of the above, but in this section

Keywords
Hölder mean log-concavity Jensen inequality triangle cevian

Citation

Stan, Aurel I.; Zapeta-Tzul, Sergio D. Log-concavity of Hölder means and an application to geometric inequalities. Involve 12 (2019), no. 4, 671--686. doi:10.2140/involve.2019.12.671. https://projecteuclid.org/euclid.involve/1559181659


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References

  • O. Bottema, R. Ž. Djordjević, R. R. Janić, D. S. Mitrinović, and P. M. Vasić, Geometric inequalities, Wolters-Noordhoff, Groningen, Netherlands, 1969.
  • P. S. Bullen, A dictionary of inequalities, Pitman Monographs Surv. Pure Appl. Math. 97, Longman, Harlow, UK, 1998.
  • P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and their inequalities, Math. Appl. (East Eur. Series) 31, Reidel, Dordrecht, 1988.
  • C. Kimberling, “Encyclopedia of triangle centers”, website, 1994, https://tinyurl.com/encytria.
  • D. S. Mitrinović, J. E. Pečarić, and V. Volenec, Recent advances in geometric inequalities, Math. Appl. (East Eur. Series) 28, Kluwer, Dordrecht, 1989.
  • G. Pólya and G. Szegő, Problems and theorems in analysis, I: Series, integral calculus, theory of functions, Grundlehren der Math. Wissenschaften 193, Springer, 1972.