※ 引述《PttFund (批踢踢基金)》之铭言： : Let p, q > 1 with 1/p + 1/q = 1. Show that : ab ≦ a^p/p + b^q/q : where a and b are nonnegative. : Note: 这个就是 Holder inequality, 然後请不要用 Young's inequality : 来证明. Another proof: It is equivalent to show that for a,b≧0, 0 < λ < 1, a^λ b^(1-λ) ≦ λa + (1-λ)b. The result is obvious if b = 0; otherwise, dividing both sides by b and setting t = a/b, we are reduced to showing that t^λ ≦ λt + (1-λ). But by elementary calculus, t^λ-λt is strictly increasing for t < 1 and strictly decreasing for t > 1, so its maximum value, namely 1-λ, occurs at t = 1. --

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