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We prove an inequality between Hilbert functions of local cohomology modulessupported in the homogeneous maximal ideal of standard graded algebras over afield, within the framework of embeddings of posets of Hilbert functions. As amain application, we prove an analogue for local cohomology of Evans'Lex-Plus-Power Conjecture for Betti numbers. This results implies some cases ofthe classical Lex-Plus-Power Conjecture, namely an inequality between extremalBetti numbers. In particular, for the classes of ideals for which theEisenbud-Green-Harris Conjecture is currently known, the projective dimensionand the Castelnuovo-Mumford regularity of a graded ideal do not decrease bypassing to the corresponding Lex-Plus-Power ideal.

The lex-plus-power inequality for local cohomology modules