實變函數論(Jensen Inequality)

Question

Let φ be a convex function on (-∞,∞) and f an integrable function on [0,1].Then∫φ(f(t))dt=φ[∫f(t)dt]書上的證明是:Let α=∫f(t)dt,and let y=m(x-α)+φ(α) be the equation of a supporting line at α.Thenφ(f(t))≧m(f(t)-α)+φ(α)最後它說兩邊對t取積分,就是我們欲證的結果,這步驟我不太懂,可否請高手解惑一下

呵呵！我真是腦筋一時轉不過來,四小時後再給你最佳解

Answer 1

因為它一開始就令α = ∫_[0,1] f(t) 了

φ(f(t)) ≧ m(f(t) - α) + φ(α) , 兩邊對 t 積分得

∫_[0,1]φ(f(t)) ≧ [∫_[0,1] m(f(t) - α)] + ∫_[0,1]φ(α)

∫_[0,1]φ(f(t)) ≧ [m∫_[0,1] (f(t) - α)] + φ(α)

∫_[0,1]φ(f(t)) ≧ {m[(∫_[0,1] (f(t)) - (∫_[0,1]α)]} + φ(α)

∫_[0,1]φ(f(t)) ≧ m(α-α) + φ(α)

∫_[0,1]φ(f(t)) ≧ φ(α) = φ(∫_[0,1] f(t))