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are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$(\int_a^b{|f(x)|^p dx})^\frac{1}{p} = f(p)$$ $$\forall p\in(a,b)$$
This question came up when thinking about $L_p$ norms as functions of $p$ and thus as a mapping of a real function to another one.

In view of the comment of Mark Meckes, I would like to know, if there are also examples of non-trivial functions, i.e. that are not identical to $1$

are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ $$\forall p\in(a,b)$$
This question came up when thinking about $L_p$ norms as functions of $p$ and thus as a mapping of a real function to another one.

In view of the comment of Mark Meckes, I would like to know, if there are also examples of non-trivial functions, i.e. that are not identical to $1$

are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$(\int_a^b{|f(x)|^p dx)})^\frac{1}{p} = f(p)$$ $$\forall p\in(a,b)$$
This question came up when thinking about $L_p$ norms as functions of $p$ and thus as a mapping of a real function to another one.

In view of the comment of Mark Meckes, I would like to know, if there are also examples of non-trivial functions, i.e. that are not identical to $1$

are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$(\int_a^b{|f(x)|^p dx})^\frac{1}{p} = f(p)$$ $$\forall p\in(a,b)$$
This question came up when thinking about $L_p$ norms as functions of $p$ and thus as a mapping of a real function to another one.

In view of the comment of Mark Meckes, I would like to know, if there are also examples of non-trivial functions, i.e. that are not identical to $1$