WOLFRAM|DEMONSTRATIONS PROJECT

Stolarsky Approximations to Popular Means

SuperscriptBox[\(n\), \(\(th\)\(\\)\)] Stolarsky

n

2.579

RadicalBox[FractionBox[\(1\ - \\*SuperscriptBox[\(x\), \(n\)]\), \(n \((1 - x)\)\)], \(n - 1\)]

other means of x and 1

maximum

RMS

average

identric

AGM

logarithmic

geometric

harmonic

minimum

0

0.2

0.4

0.6

0.8

1

x

0

0.2

0.4

0.6

0.8

1

mean(x, 1)

n = 2.579

0.2

0.4

0.6

0.8

1

x

-0.01

0

0.01

error

The Stolarsky mean provides approximations to popular means, which shows that it generalizes some of them and that these means are strictly ordered by strict inequality over

x

on

(0,1)

.