Jensen's Theorem

Author

Eric W. Weisstein
January 11, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComplexAnalysis/JensensTheorem.nb.

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Figure

<<Graphics`Colors`
<<Utilities`Typesetting`

JensenDisks[p_,z_]:=Module[{},
Map[{Re[#],Im[#]}&,List@@
({#,Conjugate[#]}&/@(Last/@Reduce[p==0&&Im[z]>0,z])),{2}]
]

circle[x__]:=Circle[Mean[{x}],Norm[Subtract@@{x}]/2]
diameter[x__]:=Line[{x}]

With[{p=z^6+z^5+z^4+z^3+z^2+z+1},
d=JensenDisks[p,z];
Show[Graphics[{
{Hue[.5],Disk@@@circle@@@d},
{Blue,circle@@@d},
PointSize[.02],
{Red,
diameter@@@d,
Point[{Re[#],Im[#]}]&/@(List@@Last/@Roots[p==0,z])
},
Point[{Re[#],Im[#]}]&/@(List@@Last/@Roots[D[p,z]==0,z])
},
AspectRatio->Automatic,Ticks->None,Axes->True,AxesFront->True,
PlotLabel->TraditionalForm[P[z]==p]
]]]

Another Version

p=z^6+z^5+z^4+z^3+z^2+z+1;

Show[Graphics[
{
{Hue[.5],Disk[{0,0},1.2]},
{Blue,Circle[{0,0},1.2]},
PointSize[.025],
Point[{Re[#],Im[#]}]&/@(List@@Last/@Roots[p==0,z])
}
],AspectRatio->Automatic]

roots=List@@(Last/@Roots[p==0,z])

p/.z->0

{
Log[Abs[(p/.z->0)((R^Length[roots])/(Times@@roots))]],
(1/(2π))Integrate[Log[Abs[p/.z->R Exp[I θ]]],{θ,0,2π}]
}

Log[((Abs[R])^6)]/.R->1.2

With[{R=1.2},(1/(2π))NIntegrate[Log[Abs[p/.z->R Exp[I θ]]],{θ,0,2π}]]