I'm examining the relationship between the expected values of the minimums of three independent, non-negative random variables $X$, $Y$, and $Z$, such that $E[2X] = E[Y] + E[Z]$. Specifically, I want to understand the relationship between $E[\min(X_1, X_2)]$ and $E[\min(Y, Z)]$, where $X_1$ and $X_2$ are independent copies of $X$.
Quick numerical experiments suggest that the latter should be smaller.
Given that expectation $E[\min(X_1, X_2)] = \int_0^\infty P(min(X_1,X_2)\geq x)dx=\int_0^\infty P(X_1\geq x)^2dx$ and $E[\min(Y, Z)] = \int_0^\infty P(min(Z,Y)\geq x)dx = \int_0^\infty P(Z\geq x)P(Y\geq x)dx$, I thought of framing this in terms of inner product spaces in $L^2$, where I compare $<X_1,X_2>$ with $<Y,Z>$, given that $<2X,1>=<Y+Z,1>$.
Are there any results from functional analysis or statistics that I can leverage for this problem?