## 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

#### Abstract

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