Logarithmic mean of two positive real numbers is well defined in the literature, it has also been extended to more than two arguments in various papers. Is there any notion of logarithmic mean of random variables or functions? Thank you for your help and time.
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You're looking for $e^{E \log X}$. It has all the nice properties you'd like it to. 


Once we know how to define a geometric mean of two quantities, the logarithmic mean is just an integral away. Indeed, say $g(t) := e^{E[(1t)\log X]+E[t\log Y]}$, then we could define a logarithmic mean as $$L(X,Y) := \int_0^1 g(t)dt.$$ Reasoning: The above idea is inspired by noting that the ordinary logarithmic mean between two positive scalars, $x$ and $y$ may be viewed as $L(x,y) = \int_0^1 x^{1t}y^tdt$, where the integrand is nothing but the (weighted) geometric mean. 

