  We prove an inequality between Hilbert functions of local cohomology modules
supported in the homogeneous maximal ideal of standard graded algebras over a
field, within the framework of embeddings of posets of Hilbert functions. As a
main application, we prove an analogue for local cohomology of Evans'
Lex-Plus-Power Conjecture for Betti numbers. This results implies some cases of
the classical Lex-Plus-Power Conjecture, namely an inequality between extremal
Betti numbers. In particular, for the classes of ideals for which the
Eisenbud-Green-Harris Conjecture is currently known, the projective dimension
and the Castelnuovo-Mumford regularity of a graded ideal do not decrease by
passing to the corresponding Lex-Plus-Power ideal.
