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Show that Im(2 + z +3z^4) <= 4 when |z|<=1
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The regular way is to go |2 + z + 3z^4| <= 2 + |z| + 3|z|^4
then we know |z|=1 so 2+1+3 =6 and we want <=4, so im missing some sort of identity or substitution .
Hopefully this is right....
Im(2 + z +3z^4)
let, z= rcosX+isinX
z^n = r^n {cos(nX)+isin(nX)} by de moivre
Im(2 + z +3z^4) = Im(2 + r{cosX+isinX} + 3r^4 cos(4X)+isin(4X))
= rsinX + 3r^4(sin{4X})
but |z|<=1, so r<=1 for all X
so Im(2 + z +3z^4) <= 1 + 3 = 4
QED