Let A,B, and X be bounded linear operators on a separable Hilbert space such that A,B are positive, X ≥ γI, for some positive real number γ, and α ∈ [0, 1]. Among other results, it is shown that if f (t) is an increasing function on [0,∞) with f (0) = 0 such that f (√ t ) is convex, then γ ∣∣∣ f (αA + (1 − α) B) + f (β |A − B|)∣∣∣ ≤ ∣∣∣α f (A) X + (1 − α) X f (B)∣∣∣ for every unitarily invariant norm, where β = min (α, 1 − α). Applications of our results are given