A mixed Holder and Minkowski inequality

Holder's inequality states that parallel to x parallel to(p)parallel to y p arallel to(q) - [x,y] greater than or equal to 0 for any (x,y) is an elemen t of L-p(Omega) x L-q(Omega) with 1/p + 1/q = 1. In the same situation we p rove the following stronger chains of inequalities, where z = y \ y \(q-2): parallel to x parallel to(p)parallel to y parallel to(q) - [x,y] greater th an or equal to (1/p) [(parallel to x parallel to(p) + parallel to z paralle l to(p))(p) - parallel to x + z parallel to(p)(p)] greater than or equal to 0 if p is an element of (1, 2], 0 less than or equal to parallel to x parallel to(p)parallel to y parallel to(q) - [x,y] less than or equal to (1/p) [(parallel to x parallel to(p) parallel to z parallel to(p))(p) - parallel to x + z parallel to(p)(p)] if p greater than or equal to 2. A similar result holds for complex valued functions with Re([x,y]) substitu ting for [x,y]. We obtain these inequalities from some stronger (though sli ghtly more involved) ones.