Abstract. 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.

Abstract. Let p = 1 be a positive real number. We determine all real numbers α = α(p) and β = β(p) such that the inequalitiesare valid for all x > 0. And, we determine all real numbers a and b such thathold for all x > 0.

A function f : (0, ∞) → R is said to be completely monotonic if (−1) n f (n) (x) ≥ 0 for all x > 0 and n = 0, 1, 2, . . .. In this paper we present several new classes of completely monotonic functions. Our functions have in common that they are defined in terms of the classical gamma, digamma, and polygamma functions. Moreover, we apply one of our monotonicity theorems to prove a new inequality for prime numbers. Some of the given results extend and complement theorems due to Bustoz & Ismail, Clark & Ismail, and other researchers.

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We present various inequalities for the digamma function, c ¼ G 0 =G, and the polygamma functions c 0 ; c 00 ; c 000 ; . . . : Our theorems improve and extend several classical and recently published results. 2000 Mathematics Subject Classification: 33B15, 26D15.