The Ramanujan Journal

, Volume 11, Issue 2, pp 225–248 | Cite as

Some classes of completely monotonic functions, II

  • Horst AlzerEmail author
  • Christian Berg


A function \(f\!:(0,\infty)\rightarrow \mathbf{R}\) is said to be completely monotonic if \((-1)^n f^{(n)}(x)\geq 0\) 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.


Complete monotonicity Gamma digamma and polygamma functions Prime numbers Inequalities 


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  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1965)zbMATHGoogle Scholar
  2. 2.
    Akhiezer, N.I.: The Classical Moment Problem and some related Questions in Analysis. English translation, Oliver and Boyd, Edinburgh (1965)zbMATHGoogle Scholar
  3. 3.
    Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comp. 66, 373–389 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alzer, H.: Mean-value inequalities for the polygamma functions. Aequat. Math. 61, 151–161 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alzer, H., Berg, C.: Some classes of completely monotonic functions. Ann. Acad. Scient. Fennicae 27, 445–460 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Anderson, G.D., Barnard, R.W., Richards, K.C., Vamanamurthy, M.K., Vuorinen, M.: Inequalities for zero-balanced hypergeometric functions. Trans. Amer. Math. Soc. 347, 1713–1723 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Andrews, G.E., Askey, R., Roy, R.: Special functions. Cambridge Univ. Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Berg, C.: Quelques remarques sur le cône de Stieltjes. In: Séminaire de Théorie du Potentiel Paris, No. 5. Lecture Notes in Mathematics 814, Springer, Berlin-Heidelberg-New York (1980)CrossRefzbMATHGoogle Scholar
  9. 9.
    Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Math. 87, Springer, Berlin (1975)CrossRefzbMATHGoogle Scholar
  10. 10.
    Berg, C., Pedersen, H.L.: A completely monotone function related to the gamma function. J. Comp. Appl. Math. 133, 219–230 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Berg, C., Pedersen, H.L.: Pick functions related to the gamma function. Rocky Mount. J. Math. 32, 507–525 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bochner, S.: Harmonic Analysis and The Theory of Probability. Univ. of California Press, Berkeley-Los Angeles (1960)zbMATHGoogle Scholar
  13. 13.
    Bondesson, L.: Generalized gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics 76, Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  14. 14.
    Bonse, H.: Über eine bekannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung. Arch. Math. Phys. 12, 292–295 (1907)zbMATHGoogle Scholar
  15. 15.
    Bustoz, J., Ismail, M.E.H.: On gamma function inequalities. Math. Comp. 47, 659–667 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Canfield, E.R.: Problem 10310. Amer. Math. Monthly 100, 499 (1993); 103, 431–432 (1996)Google Scholar
  17. 17.
    Clark, W.E., Ismail, M.E.H.: Inequalities involving gamma and psi functions. Anal. Appl. 1, 129–140 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Czinder, P., Páles, Z.: A general Minkowski-type inequality for two variable Gini means. Publ. Math. Debrecen 57, 203–216 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dang, H., Weerakkody, G.: Bounds for the maximum likelihood estimates in two-parameter gamma distribution. J. Math. Anal. Appl.245, 1–6 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Donoghue, Jr., W.F.: Monotone Matrix Functions and Analytic Continuation. Springer, Berlin (1974)CrossRefzbMATHGoogle Scholar
  21. 21.
    Dubourdieu, J.: Sur un théorème de M.S. Bernstein relatif à la transformation de Laplace-Stieltjes. Compositio Math. 7, 96–111 (1939)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Erdélyi, A. (ed.): Higher Transcendental Functions, Vol. 1. McGraw-Hill, New York (1953)zbMATHGoogle Scholar
  23. 23.
    Fink, A.M.: Kolmogorov-Landau inequalities for monotone functions. J. Math. Anal. Appl. 90, 251–258 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gautschi, W.: The incomplete gamma function since Tricomi. In: Tricomi's ideas and contemporary applied mathematics. Atti Convegni Lincei 147, pp. 207–237, Accad. Naz. Lincei, Rome (1998)MathSciNetGoogle Scholar
  25. 25.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge Univ. Press, Cambridge (1952)zbMATHGoogle Scholar
  26. 26.
    Hirsch, F.: Transformation de Stieltjes et fonctions opérant sur les potentiels abstraits. In: Lecture Notes in Mathematics 404, 149–163, Springer, Berlin-Heidelberg-New York (1974)CrossRefzbMATHGoogle Scholar
  27. 27.
    Ismail, M.E.H., Lorch, L., Muldoon, M.E.: Completely monotonic functions associated with the gamma function and its q-analogues. J. Math. Anal. Appl. 116, 1–9 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Itô, M.: Sur les cônes convexes de Riesz et les noyaux complètement sous-harmoniques. Nagoya Math. J. 55, 111–144 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kershaw, D., Laforgia, A.: Monotonicity results for the gamma function. Atti Accad. Sci. Torino 119, 127–133 (1985)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Kimberling, C.H.: A probabilistic interpretation of complete monotonicity. Aequat. Math. 10, 152–164 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Merkle, M.: On log-convexity of a ratio of gamma functions. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 8, 114–119 (1997)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Mitrinović, D.S., Sándor, J., Crstici, B.: Handbook of Number Theory. Kluwer, Dordrecht (1996)zbMATHGoogle Scholar
  33. 33.
    Muldoon, M.E.: Some monotonicity properties and characterizations of the gamma function. Aequat. Math. 18, 54–63 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Palumbo, B.: Determinantal inequalities for the psi function. Math. Inequal. Appl. 2, 223–231 (1999)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Panaitopol, L.: An inequality involving prime numbers. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11, 33–35 (2000)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Reuter, G.E.H.: ÜUber eine Volterrasche Integralgleichung mit totalmonotonem Kern. Arch. Math. 7, 59–66 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Robin, G.: Estimation de la fonction de Tchebychef θ sur le k-ième nombre premier et grandes valeurs de la fonction ω nombre de diviseurs premiers de n. Acta Arith. 42, 367–389 (1983)MathSciNetGoogle Scholar
  38. 38.
    Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Illinois J. Math. 6, 64–94 (1962)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Sándor, J.: Sur la fonction gamma. Publ. C.R.M.P. Neuchòtel Série I, 21, 4–7 (1989)zbMATHGoogle Scholar
  40. 40.
    Sándor, J.; On some diophantine equations involving the factorial of a function. Seminar Arghiriade 21, 1–4 (1989) Univ. Timisoara (Romania)Google Scholar
  41. 41.
    Thorin, O.: On the infinite divisibility of the Pareto distribution. Scand. Actuarial. J., 31–40 (1977)Google Scholar
  42. 42.
    Thorin, O.: On the infinite divisibility of the lognormal distribution. Scand. Actuarial. J., 121–148 (1977)Google Scholar
  43. 43.
    Trimble, S.Y., Wells, J., Wright, F.T.: Superadditive functions and a statistical application. SIAM J. Math. Anal. 20, 1255–1259 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Widder, D.V.: The Laplace Transform. Princeton Univ. Press, Princeton (1941)zbMATHGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.WaldbrölGermany
  2. 2.Department of MathematicsUniversity of CopenhagenDenmark

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