On some inequalities for the gamma and psi functions

We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, star-shaped, and superadditive functions which are related to Γ and ψ. 1 Euler’s gamma function Γ(x) = ∫ ∞ 0 e−ttx−1 dt (x > 0) is one of the most important functions in analysis and its applications. The history and the development of this function are described in detail in a paper by P. J. Davis [10]. There exists a very extensive literature on the gamma function. In particular, numerous remarkable inequalities involving Γ and its logarithmic derivative ψ = Γ′/Γ have been published by different authors; see, e.g., [2], [3], [6], [7], [9], [12], [13], [18]–[27], [29]–[33], [35]–[46], [50]. Many of these inequalities follow immediately from the monotonicity properties of functions which are closely related to Γ and ψ. In several recent papers [2], [9], [24], [39] it is proved that these functions are not only monotonic, but even completely monotonic. We recall that a function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I which alternate successively in sign, that is, (−1)f (x) ≥ 0 (1.1) for all x ∈ I and for all n ≥ 0. If inequality (1.1) is strict for all x ∈ I and all n ≥ 0, then f is said to be strictly completely monotonic. It is known that completely monotonic functions play an eminent role in areas like probability theory [15], numerical analysis [49], physics [11], and the theory of special functions. For instance, M. E. Muldoon [39] showed how the notation of complete monotonicity can be used to characterize the gamma function. An interesting exposition of the main results on completely monotonic functions is given in [48]. “In view of the importance of completely monotonic functions . . . it may be of interest to add to the available list of such functions” [24, p. 1]. It is the main Received by the editor October 13, 1995 and, in revised form, March 4, 1996. 1991 Mathematics Subject Classification. Primary 33B15; Secondary 26D07.