A mixed hölder and minkowski inequality

Hölder's inequality states that ∥x∥_p ∥y∥_q - 〈z, y〉 ≥ 0 for any (x, y) ∈ £^p(Ω) × £^q(Ω) with 1/p + 1/g = 1. In the same situation we prove the following stronger chains of inequalities, where z = y|y|^q-2: ∥x∥_p ∥y∥_q, - 〈x, y〉 ≥ (1/P)[(∥x∥_p + ∥z∥_p)^p - ∥x + z∥^p_p] ≥ 0 if P ∈ (1,2], 0 ≤ ∥x∥_p ∥y∥_q - 〈x, y〉 ≤ (1/p) [(∥x∥_p + ∥z∥_p)^p - ∥x + z∥^p_p] if p ≥ 2. A similar result holds for complex valued functions with Re(〈x, y〉) substituting for 〈x,y〉. We obtain these inequalities from some stronger (though slightly more involved) ones.