※ 引述《u2150260 (鴻哥)》之銘言：
: 1. Find the necessary and sufficient conditions on complex number z and w
: in order that |z+w|=|z|+|w|
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|z+w|^2 = (z+w)(z+w) = |z|^2 + |w|^2 + z w + w z
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= |z|^2 + |w|^2 + 2 Re( z w)
≦ |z|^2 + |w|^2 + 2|z||w| = (|z|+|w|)^2
因
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Re( z w) = Re[(a-bi)(c+di)] = ac + bd ≦ |(ac + bd)|
≦ √(a^2 + b^2)√(c^2 + d^2) = |z||w|
等號成立條件a/c = b/d
所以是當z = tw for t belongs to postive number
: 2.For any real number z and w ,prove that
: |z|/(1+|z|) + |w|/(1+|w|) ≧ |z+w|/(1+|z+w|)
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◆ From: 122.124.107.161
※ 編輯: Honor1984 來自: 122.124.107.161 (07/04 22:18)