※ 引述《Honor1984 (希望願望成真)》之銘言：
: ※ 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|
: ─── ─ ─
: |z+w|^2 = (z+w)(z+w) = |z|^2 + |w|^2 + z w + w z
: ─
: = |z|^2 + |w|^2 + 2 Re( z w)
: ≦ |z|^2 + |w|^2 + 2|z||w| = (|z|+|w|)^2
: 因
: ─
: 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|)
Prove z and w are positive
z/(1+z) + w/(1+w) >= z/(1+z+w) + w/(1+z+w)
= (z+w)/(1+z+w)
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