[...]... 2 βi B2 i=1 1 + 2B 2 2 αi i=1 n 2 βi = i=1 1 2 1 A + B2 = 1 2A2 2B 2 Thus, 1 AB n n αi βi ≤ 1 ⇒ i=1 1 2 n αi βi ≤ 2 αi i=1 i=1 1 2 n 2 βi i=1 Equality holds if αi = kβi , for i = 1, , n ∗ ∗ 2) The second equation is trivial since |xi yi | = |xi ||yi | To see this write xi and yi in polar jθxi jθyi ∗ coordinates as xi = ρxi e and yi = ρyi e Then, |xi yi | = |ρxi ρyi ej(θxi −θyi ) | = ρxi ρyi = |xi .