17.3. UNIFORM CONVEXITY OF Lp 453

Lemma 17.3.4 For any p≥ 2 the following inequality holds for any t ∈ [0,1] ,∣∣∣∣1+ t2

∣∣∣∣p + ∣∣∣∣1− t2

∣∣∣∣p ≤ 12(|t|p +1)

Proof: It is clear that, since p ≥ 2, the inequality holds for t = 0 and t = 1.Thus itsuffices to consider only t ∈ (0,1). Let x = 1/t. Then, dividing by 1/t p, the inequalityholds if and only if (

x+12

)p

+

(x−1

2

)p

≤ 12(1+ xp)

for all x≥ 1. Let

f (x) =12(1+ xp)−

((x+1

2

)p

+

(x−1

2

)p)Then f (1) = 0 and

f ′ (x) =p2

xp−1−

(p2

(x+1

2

)p−1

+p2

(x−1

2

)p−1)

Since p−1≥ 1, by convexity of f (x) = xp−1,

f ′ (x)≥ p2

xp−1− p

(x+1

2 + x−12

2

)p−1

=p2

xp−1− p( x

2

)p−1≥ 0

Hence f (x)≥ 0 for all x≥ 1.

Corollary 17.3.5 If z,w ∈ C and p≥ 2, then∣∣∣∣ z+w2

∣∣∣∣p + ∣∣∣∣ z−w2

∣∣∣∣p ≤ 12(|z|p + |w|p) (17.3.7)

Proof: One of |w| , |z| is larger. Say |z| ≥ |w| . Then dividing both sides of the proposedinequality by |z|p it suffices to verify that for all complex t having |t| ≤ 1,∣∣∣∣1+ t

2

∣∣∣∣p + ∣∣∣∣1− t2

∣∣∣∣p ≤ 12(|t|p +1)

Say t = reiθ where r ≤ 1.Then consider the expression∣∣∣∣1+ reiθ

2

∣∣∣∣p + ∣∣∣∣1− reiθ

2

∣∣∣∣pIt is 2−p times (

(1+ r cosθ)2 + r2 sin2 (θ))p/2

+((1− r cosθ)2 + r2 sin2 (θ)

)p/2

=(1+ r2 +2r cosθ

)p/2+(1+ r2−2r cosθ

)p/2,