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For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$. Let $0 < a < b < c < \infty$ and prove the following: $$ \|f\|_b \leqslant \max\{\|f\|_a, \|f\|_c\}. $$ Any help is appreciated because I dont understand the solution underneath

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    The idea here is that $||f||_p \leq ||f||_q$ if $1 \leq p \leq q \leq \infty$ . But the inequality reverses in $0 . Hope this helps.2012-11-19
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    That is to say $||f||_1$ is a point of minima.2012-11-19
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    I think here we have to use Holder inequality in some way , so can you give more details please ? So do I have to take cases where a>1 and when a<1 with b,c > 1 ?2012-11-19

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