Reciprocally convex functions

We say that f is reciprocally convex if x → f (x) is concave and x → f (1/x) is convex on (0,+∞). Reciprocally convex functions generate a sequence of quasi-arithmetic means, with the first one between harmonic and arithmetic mean and others above the arithmetic mean. We present several examples related to the gamma function and we show that if f is a Stieltjes transform, then −f is reciprocally convex. An application in probability is also presented.  2004 Elsevier Inc. All rights reserved.