We discuss when law-invariant convex functionals “collapse to the mean”. More precisely, we show that, in a large class of spaces random variables and under mild semicontinuity assumptions, expectation functional is, up an affine transformation, only that is linear along direction nonconstant variable with nonzero expectation. This extends results obtained literature bounded setting additional ...

We show that, for a (not necessarily continuous) weakly contractive mean-type mapping $\mathbf{M} \colon I^p\to I^p$ (where $I$ is an interval and $p \in \mathbb{N}$), the functional equation $K \circ \mathbf{M}=K$ has at most one solution in family of continuous means I^p \to I$. Some general approach to latter also given.

We give conditions on the functions σ and u on R such that if u is given by the convolution of σ and u, then u is harmonic on R.

A positive real interval, [a, b] can be partitioned into sub-intervals such that sub-interval widths divided by sub-interval ”‘average”’ values remains constant. That both Arithmetic Mean and Geometric Mean ”‘average”’ values produce constant ratios for the same log scale is the stated invariance proved in this short note. The continuous analog is briefly considered and shown to have similar pr...