 

Z_{Atkinson} = 1Z_{MacRae[1]} ≥ 1exp(Σ_{i=1..N}(E_{i}*ln(A_{i}/E_{i}))/E_{total})*E_{total}/A_{total} 
nosniktA inequality 
Z_{nosniktA} ≥ 1exp(Σ_{i=1..N}(A_{i}*ln(E_{i}/A_{i}))/A_{total})*A_{total}/E_{total} 
TheilT redundancy 
R_{Theil} 
= ln(1Z_{Atkinson}) = ln(Z_{MacRae}) 

≥ ln(A_{total}/E_{total})  Σ_{i=1..N}(E_{i}*ln(A_{i}/E_{i}))/E_{total} 

TheilL redundancy 
R_{liehT} 
= ln(1Z_{nosniktA}) 

≥ ln(E_{total}/A_{total})  Σ_{i=1..N}(A_{i}*ln(E_{i}/A_{i}))/A_{total} 

TheilS redundancy Symmetric redundancy 
 
R_{sym} 
= ln(1Z_{sym}) = 2*Z_{Plato}*artanh(Z_{Plato}) 

= (R_{Theil}(EA)+R_{Theil}(AE))/2 = (R_{Theil}+R_{liehT})/2 

≥ Σ_{i=1..N}(ln(E_{i}/A_{i})*(E_{i}/E_{total}A_{i}/A_{total}))/2 

Symmetric inequality 
Z_{sym} 
= 1exp(R_{sym}) = 1√((1Z_{Atkinson})*(1Z_{nosniktA})) 

≥ 1exp(Σ_{i=1..N}(ln(A_{i}/E_{i})*(E_{i}/E_{total}A_{i}/A_{total}))/2) 

Hoover inequality 
Z_{Hoover} ≥ Σ_{i=1..N}E_{i}/E_{total}A_{i}/A_{total}/2 
Coulter inequality 
Z_{Coulter} ≥ √(Σ_{i=1..N}(E_{i}/E_{total}A_{i}/A_{total})^{2}/2) 
Gini inequality 
sort data: E_{i}/A_{i}>E_{i1}/A_{i1}
Z_{Gini} ≥ 1Σ_{i=1..N}((2*Σ_{k=1..i}(E_{k})E_{i})*A_{i})/(E_{total}*A_{total})

EU inequality 
1:a = (1Z_{Gini})/(1+Z_{Gini}) is the SOEP "equality parameter"
therefore: Z_{Europe} = 2*Z_{Gini}/(1+Z_{Gini})

Plato inequality 
inverse functions:
Z_{sym}
= 1((1Z_{Plato})/(1+Z_{Plato}))^{ZPlato}
R_{sym}
= 2*Z_{Plato}*artanh(Z_{Plato})
approximation:
Z_{Plato} ≈ 1  arcsin((1Z_{sym})^{(0.06*Zsym+0.61)})*2/π,
error < 0.002 for Z_{sym} < 0.75
fast recursion (there is a better version in onOEI1.0.5.py):
initialize: Z_{Plato} ≈ 1  arcsin(exp(R_{sym}(0.06/exp(R_{sym})0.67)))*2/π
repeat:
Z_{last} = Z_{Plato}
Z_{Plato} = tanh(R_{sym}/(Z_{Plato}+Z_{last}))
until
2*Z_{Plato}*artanh(Z_{Plato})  R_{sym} is small enough.
format for comparison to the "Pareto Principle":
a : b = (1+Z_{Plato})/2 : (1Z_{Plato})/2
Z_{Plato} = 2a1 = 2b1

Redistributive Aggression 
R_{A} 
= (R_{Theil}+R_{liehT})/2  Z_{Hoover} = R_{sym}  Z_{Hoover} 

= Σ_{i=1..N}(ln(E_{i}/A_{i})*(E_{i}/E_{total}A_{i}/A_{total})  E_{i}/E_{total}A_{i}/A_{total})/2


 