Inequalities for Gamma Function Ratios

Write R(x, y) = Γ(x + y) Γ(x). Inequalities for this ratio have interesting applications, and have been considered by a number of writers over a long period. In a Monthly article [7], Wendel showed that x(x + y) y−1 ≤ R(x, y) ≤ x y for 0 ≤ y ≤ 1. (1) Wendel's method was an ingenious application of Hölder's inequality to the integral definition of the gamma function. Note that both inequalities are exact when y is 0 or 1, since R(x, 0) = 1 and R(x, 1) = x. Without reference to Wendel, and by quite different methods, Gautschi [4] proved x(x + 1) y−1 ≤ R(x, y) ≤ x y for 0 ≤ y ≤ 1. (2) The lower bound in (2) is weaker than the one in (1), and is not exact when y = 0. Gautschi only stated the result for integer values of x, but his method does not require this. He also established the more elaborate lower bound x exp[(y − 1)ψ(x + 1)], where ψ(x) = Γ (x)/Γ(x), which implies (2) because ψ(x) < log x. Refinements have appeared in many later articles, e.g. [3, 5, 6]. Most of them take the form of expressions in terms of ψ(x), but one of Kershaw's bounds is x(x + y 2) y−1. Here we confine attention to bounds of the simple type seen in (1) and (2). Artin's classic book [2] was published in 1931, long before either Wendel or Gautschi. In it (p. 14) we find the statement (x − 1) y ≤ R(x, y) ≤ x y for 0 ≤ y ≤ 1, (3) (actually only stated for integers x). Again, the lower bound is weaker than the one in (1), though this is not quite so transparent, and is not exact at 1. Artin did not state (3) as a result in its own right, but only as a step in the proof of another theorem. Perhaps for this reason, his result appears to have been overlooked by most later writers, including Wendel and Gautschi. Indeed, inequalities of this type have generally been referred to as " Gautschi-type inequalities, " with scant respect to either Artin or Wendel. As well as having appeared earlier, Artin's method is much simpler than those of the later writers-indeed, as we shall see, it makes the result seem almost trivial! The only 1