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Abstract
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The log-concavity of the Hölder mean of two numbers, as a function of its index, is presented first.
The notion of
-cevian
of a triangle is introduced next, for any real number
.
We use this property of the Hölder mean to find the smallest index
such that the length of an
-cevian of a triangle is less
than or equal to the
-Hölder
mean of the lengths of the two sides of the triangle that are adjacent to that
cevian.
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Keywords
Hölder mean, log-concavity, Jensen inequality, triangle,
cevian
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Mathematical Subject Classification 2010
Primary: 26A06, 26D99
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Milestones
Received: 23 May 2018
Revised: 9 November 2018
Accepted: 15 November 2018
Published: 16 April 2019
Communicated by Sever S. Dragomir
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