Question: Let belong to both and , with . Show that for all .

There is a pretty neat trick to do this question, known as the “interpolation technique”. The proof is as follows.

For , there exists such that . This is the key “interpolation step”. Once we have this, everything flows smoothly with the help of Holder’s inequality.

Thus .

Note that the magical thing about the interpolation technique is that and are Holder conjugates, since is easily verified.

Undergraduate Math Books

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