Wilker's Inequalities

Author

Eric W. Weisstein
July 12, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/WilkersInequalities.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

eqns={2+(2/45)(x^3)Tan[x],(((Sin[x])^2)/(x^2))+((Tan[x])/x),2+(16/(π^4))(x^3)Tan[x]};

Inequality

Greater@@eqns

Resolve[ForAll[x,
0<x<π/2,
Greater@@eqns
],Reals]

Reduce[Greater@@eqns&&0<x<π/2,x]//Timing

First Inequality

FullSimplify[-Subtract@@Take[eqns,2],0<x<π/2]

FullSimplify[D[-2+(((Sin[x])^2)/(x^2))+((1/x)-((2 (x^3))/45)) Tan[x],x]]

Refine[ForAll[x,
0<x<Pi/2,
-2+(((Sin[x])^2)/(x^2))+((1/x)-((2 (x^3))/45)) Tan[x]>0]]

Reduce[-45+180 (x^2)-8 (x^6)+45 Cos[4 x]-12 (x^5) Sin[2 x]+45 x Sin[4 x]==0&&Inequality[0,Less,x,Less,Pi/2],x,Reals]//Timing

Reduce[-45+180 (x^2)-8 (x^6)+45 Cos[4 x]-12 (x^5) Sin[2 x]+45 x Sin[4 x]>0&&Inequality[0,Less,x,Less,Pi/2],x,Reals]//Timing

Minimize[{-Subtract@@Take[eqns,2],0<x<π/2},x]//Timing

NMinimize[{-Subtract@@Take[eqns,2],0<x<π/2},x]//Timing

Second Inequality

Minimize[{-Subtract@@Take[eqns,-2],0<x<π/2},x]//Timing

Plot

<<Graphics`Colors`

Drop[eqns,{2}]-eqns[[2]]

Plot[Evaluate[Drop[eqns,{2}]-eqns[[2]]],{x,0,π/2},PlotStyle->{Red,Blue},AxesLabel->TraditionalForm/@{x,(Subscript[f, n])-(Subscript[f, 2])},PlotRange->{-.05,.005}]