EDB — 10F

[10F] Prerequisites:[1H8]↺↻.Having fixed \(t,s∈[1,∞]\) with \(s{\gt}t\) and \(x∈ℝ^ n\), show that \(\| x\| _{s}≤ \| x\| _{t}\). Also show that \(\| x\| _{s}{\lt} \| x\| _{t}\) if \(n≥ 2\) and \(x≠ 0\) and \(x\) is not a multiple of one of the vectors of the canonical basis \(e_ 1,\ldots e_ n\).

Hints:

1. use that \(1+t^ p≤ (1+t)^ p\) for \(p≥ 1\) and \(t≥ 0\); or

2. use Lagrange multipliers; or

3. remember that \(f(a+b){\gt} f(a)+f(b)\) when \(a≥ 0, b{\gt}0\) \(f(0)=0\) and \(f:[0,∞)→ℝ\) is strictly convex and continuous in 0 (see exercise [192]↺↻), therefore derive \(\frac{d\hskip5.5pt}{d{t}}(\log \| x\| _{t})\) and set \(f(z)=z\log (z)\)).