Abi-Khuzam Inequality

Author

Eric W. Weisstein
March 26, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Trigonometry/Abi-KhuzamInequality.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Abi-KhuzamInequality.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Analytic Check

Resolve[ForAll[{a1,a2,a3},
a1+a2+a3==π&&0<a1<π&&0<a2<π&&0<a3<π,
Sin[a1]Sin[a2]Sin[a3]≤(((((3Sqrt[3])/(2π))))^3)a1 a2 a3
],Reals]

N[{(((((3Sqrt[3])/(2π))))^3),π/3},20]

With[{ϵ=10^-6},NMaximize[{((Sin[a1]Sin[a2]Sin[a3])/(a1 a2 a3)),a1+a2+a3==π&&ϵ<a1<π&&ϵ<a2<π&&ϵ<a3<π},{a1,a2,a3},WorkingPrecision->20]]

((Sin[a1]Sin[a2]Sin[a3])/(a1 a2 a3))/.Thread[{a1,a2,a3}->π/3]

NMaximize[{((Sin[a1]Sin[a2]Sin[a3])/(a1 a2 a3)),a1+a2+a3==π},{a1,a2,a3},WorkingPrecision->20]

Maximize[{((Sin[a1]Sin[a2]Sin[a3])/(a1 a2 a3)),a1+a2+a3==π},{a1,a2,a3}]

With[{a3=π-a1-a2},Maximize[((Sin[a1]Sin[a2]Sin[a3])/(a1 a2 a3)),{a1,a2}]]

With[{a3=π-a1-a2},Maximize[{((Sin[a1]Sin[a2]Sin[a3])/(a1 a2 a3)),0<a1<π&&0<a2<π},{a1,a2}]]

Numerical Check

Max@Table[Times@@(Sin[#]/#&/@{r2=Random[Real,{0,π-(r1=Random[Real,{0,π}])}],r1,π-r1-r2}),{50000}]//Timing

Plot of Distribution

<<Graphics`Graphics`
<<Statistics`DataManipulation`

BarChart[BinCounts[Table[Times@@(Sin[#]/#&/@{r2=Random[Real,{0,π-(r1=Random[Real,{0,π}])}],r1,π-r1-r2}),{100000}],{0,(((((3.Sqrt[3])/(2π))))^3),.01}],BarGroupSpacing->0]//Timing