Vol. 12, No. 4, 2019

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

Aurel I. Stan and Sergio D. Zapeta-Tzul

Vol. 12 (2019), No. 4, 671–686
DOI: 10.2140/involve.2019.12.671
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.

Keywords
Hölder mean, log-concavity, Jensen inequality, triangle, cevian
Mathematical Subject Classification 2010
Primary: 26A06, 26D99
Milestones
Received: 23 May 2018
Revised: 9 November 2018
Accepted: 15 November 2018
Published: 16 April 2019

Communicated by Sever S. Dragomir
Authors
Aurel I. Stan
Department of Mathematics
Ohio State University at Marion
Marion, OH
United States
Sergio D. Zapeta-Tzul
Department of Mathematics
Universidad del Valle de Guatemala
Guatemala
Guatemala