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Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then

$$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$

I refer to $(*)$ as the Liapunov's Inequality for $L_p$ spaces. I've been told that it is possible to prove this inequality by "an appropiate application" of the regular Hölder inequality (that is, $||fg||_1 \leq ||f||_p ||g||_q$, where $p$ and $q$ are conjugate exponents). I know how to prove $(*)$ using other inequalities closely related to the regular Hölder inequality, but not that inequality itself. After awhile playing with the regular Hölder inequality, I haven't been able to get $(*)$. So, can someone illuminate me with the "appropiate application" of the regular Hölder inequality?

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up vote 5 down vote accepted

Write $$\int |f|^rd\mu=\int\color{green}{|f|^{p\lambda}}\color{red}{|f|^{q(1-\lambda)}}d\mu,$$ then apply Hölder's inequality to the exponent $\frac 1{\lambda}>1$ (its conjugate exponent is $\frac 1{1-\lambda}$). This gives $$\int |f|^rd\mu\leqslant \color{green}{\left(\int |f|^p\right)^{\lambda}}\color{red}{\left(\int |f|^q\right)^{1-\lambda}}=\color{green}{\lVert f\rVert_p^{p\lambda}}\color{red}{\lVert f\rVert_q^{q(1-\lambda)}},$$ what is wanted.

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Thank you! Indeed, it was really simple. –  ragrigg Nov 11 '12 at 22:17
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