EDB — 10M

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E19

[10M] Let be given \(p,q∈[1,∞]\) such that \(1/p + 1/q = 1\)  1 and \(x,y∈ℝ^ n\); show the Hölder inequality in this form

\begin{equation} ∑_{i=1}^ n|x_ i y_ i| ≤ \| x\| _ p \| y\| _ q\quad .\label{eq:dis_ Holder_ val_ ass} \end{equation}
20

In what cases is there equality?

Tips: Fix \(x_ i,y_ i≥ 0\). For the cases with \(p,q{\lt}∞\) you can:

  • use Young inequality ([194] or [1V7]);

  • use Lagrange multipliers;

  • start from the case \(n=2\) and set \(x_ 2=t x_ 1\) and \(y_ 2=a y_ 1\); then, for cases \(n≥ 3\) use induction.

Solution 1

[10N]

  1. This means that if \(p=1\) then \(q=∞\) ; and vice versa.
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Authors: "Mennucci , Andrea C. G." .
Bibliography
Book index
  • normed vector space
  • Young inequality
  • Hölder inequality
  • Lagrange multiplier
  • \( \Vert \cdot \Vert _p\) , in \( ℝ ^n\)
  • \( \Vert \cdot \Vert _\infty \) , in \( ℝ ^n\)
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