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单词 ConvexityOfTangentFunction
释义

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 thata continuousMathworldPlanetmath real function f is convex (http://planetmath.org/ConvexFunction) 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 thearithmetic-geometric mean inequality (http://planetmath.org/ArithmeticGeometricMeansInequality),

-2xy-x2-y2
1-2xy+x2y21-x2-y2+x2y2
(1-xy)2(1-x2)(1-y2),

so

(1-xy)2(1-x2)(1-y2)1.

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.

Hence,

12(tan(2u)+tan(2v))=x+y-x2y-xy2(1-x2)(1-y2)
=(x+y)(1-xy)(1-x2)(1-y2)
=x+y1-xy(1-xy)2(1-x2)(1-y2)
x+y1-xy=tan(u+v),

so the tangent function is convex.

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更新时间:2025/5/4 10:08:48