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Let $f(x)$ be a non-negative function. Is there an upper bound for the following quantity?

$$\mathbb{E}\left[f(x)^{1-ab} \right] \mathbb{E}\left[f(x)^{a} \right]^b \leq \, ??,$$

where $a > 0$ and $b \in [0, 1]$. While I am aware that Hölder's inequality provides a lower bound, I am seeking an upper bound.

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1 Answer 1

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Let $X$ be a coin flip, set $f$ to $x\mapsto |x|$. Then $$ E[f(X)^{1-ab}] = E[(\frac{1}{f(X)})^{ab-1}] = [\text{Assume: } ab>1]=\frac{1}{2}\sum_{j=0}^1\frac{1}{|j|^{ab-1}}=\infty. $$

This example is not an especially bad case since $E[f(X)]<\infty$ so $E[f(X)]^{1-ab}$ and other variations (that could be a factor in the upper bound) are also finite. Another point indicating that stronger assumptions are needed: Setting $x\mapsto 1/g(x)$ gives $$ E[f(X)^{1-ab}] = E[(\frac{1}{f(X)})^{ab-1}]= E[g(X)^{ab-1}]. $$

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