Math Help - A characterization of an almost everywhere constant variable

I've been thinking for several hours now, and can't see it.

Let be a probability space. Let be a -integrable variable (in the Lebesgue sense). Then the following conditions are equivalent:

- is -almost everywhere constant
-

Obviously I must be missing some inequality, probably well known. Any help will be appreciated.

This is an attainment of Jensen's Inequality.

Thank you very much.