Mills' ratio: Reciprocal concavity and functional inequalities
Árpád Baricz · arXiv (Cornell University) · 2010
AbstractThis note contains sufficient conditions for the probability density function of an arbitrary continuousunivariate distribution such that the corresponding Mills ratio to be reciprocally convex (concave). Toillustrate the applications of the main results, the Mills ratio of some common continuous univariate dis-tributions, like gamma, log-normal and Student’s t distributions, are discussed in details. The applicationto monopoly theory is also summarized. Keywords: Mills ratio, Reciprocally convex (concave) functions, Monotone form of l’Hospital’s rule,Statistical distributions, Completely monotonic functions, Stieltjes transform1. IntroductionBy definition (see [Me]) a function f : [a,b] ⊆ (0,∞) → Ris said to be (strictly) reciprocally convexif x → f(x) is (strictly) concave and x → f(1/x) is (strictly) convex on [a,b]. Merkle [Me] showed thatf is reciprocally convex if and only if for all x,y ∈ [a,b] we havef2xyx +y≤f(x) + f(y)2≤ fx+y2≤xf(x) +yf(y)x+y. (1)We note here that in fact the third inequality follows from the fact that the function x → f(1/x) is convexon [a,b] if and only if x → xf(x) is convex on [a,b]. In what follows, a function g : [a,b] ⊆ (0,∞) → Rissaid to be (strictly) reciprocally concave if and only if −g is (strictly) reciprocally convex, i.e. if x → g(x)is (strictly) convex and x → g(1/x) is (strictly) concave on [a,b]. Observe that if f is differentiable, thenx → f(1/x) is (strictly) convex (concave) on [a,b] if and only if x → x