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We prove the following suband superadditive properties of the psi function. (i) The inequality ψ ( (x+ y) ) ≤ ψ(x) + ψ(y) (α ∈ R) holds for all x, y > 0 if and only if α ≤ α0 = −1.0266 . . . . Here, α0 is given by 20 = inf t>0 ψ−1 ( 2ψ(t) )

Sub- and super-additive properties of the psi function