convexity of tangent function

We will show that the tangent function is convex on the interval [0,π/2). To do this, we will use the addition formulaPlanetmathPlanetmath for the tangentPlanetmathPlanetmathPlanetmath and the fact that a continuousMathworldPlanetmath real function f is convex ( if and only if f((x+y)/2)(f(x)+f(y))/2.

We start with the observation that, if 0x<1 and 0y<1, then by the arithmetic-geometric mean inequality (,

-2xy -x2-y2
1-2xy+x2y2 1-x2-y2+x2y2
(1-xy)2 (1-x2)(1-y2),



Let u and v be two numbers in the interval [0,π/4). Set x=tanu and y=tanv. Then 0x<1 and 0y<1. By the addition formula, we have

tan(2u) =2x1-x2
tan(u+v) =x+y1-xy
tan(2v) =2y1-y2.


12(tan(2u)+tan(2v)) =x+y-x2y-xy2(1-x2)(1-y2)

so the tangent function is convex.

Title convexity of tangent function
Canonical name ConvexityOfTangentFunction
Date of creation 2013-03-22 17:00:12
Last modified on 2013-03-22 17:00:12
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 14
Author rspuzio (6075)
Entry type Result
Classification msc 26A09