Sharp inequalities for the digamma and polygamma functions

Forum Mathematicum

Volumn 16, Issue 2, 2004, Pages 181-221

  Alzer, Horst ^a  

^a NONE   (Germany)

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[No Author keywords available]

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EID: 1042279113     PISSN: 09337741     EISSN: None     Source Type: Journal    
DOI: 10.1515/form.2004.009     Document Type: Article

References (43)

1. Abramowitz M. and Stegun I. A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York 1965
2. Alzer H.: On some inequalities for the gamma and psi functions. Math. Comp. 66 (1997), 373-389
3. Alzer H.: Inequalities for the gamma and polygamma functions. Abh. Math. Sem. Hamburg 68 (1998), 363-372
4. n. J. Math. Anal. Appl. 252 (2000), 353-363
5. Alzer H.: Mean-value inequalities for the polygamma functions. Aequationes Math. 61 (2001), 151-161
6. Alzer H. and Ruehr O. G.: A submultiplicative property of the psi function. J. Comp. Appl. Math. 101 (1999), 53-60
7. Alzer H. and Wells J.: Inequalities for the polygamma functions. SIAM J. Math. Anal. 29 (1998), 1459-1466
8. Anderson G. D., Barnard R. W., Richards K. C., Vamanamurthy M. K., and Vuorinen M.: Inequalities for zero-balanced hypergeometric functions. Trans. Amer. Math. Soc. 347 (1995), 1713-1723
9. Berndt B. C.: Ramanujan's Notebook, Part IV. Springer, New York 1994
10. Borgwardt K. H.: The Simplex Method. Springer, Berlin 1987
11. Bullen P. S., Mitrinović D. S., and Vasić P. M.: Means and Their Inequalities. Reidel, Dordrecht 1988
12. Cvijović D. and Klinowski J.: Inequalities for the Landau constants. Math. Slovaca 50 (2000), 159-164
13. Dang H. and Weerakkody G.: Bounds for the maximum likelihood estimates in two-parameter gamma distribution. J. Math. Anal. Appl. 245 (2000), 1-6
14. Elezović N., Giordano C., and Pečarić J.: The best bounds in Gautschi's inequality. Math. Inequal. Appl. 3 (2000), 239-252
15. English B. J. and Rousseau G.: Bounds for certain harmonic sums. J. Math. Anal. Appl. 206 (1997), 428-441
16. Gautschi W.: The incomplete gamma functions since Tricomi. In: Tricomi's Ideas and Contemporary Applied Mathematics. Accademia Nazionale Dei Lincei, Rome 1998, 203-237
17. Gini C.: Di una formula comprensiva. Metron 13 (1938), 3-22
18. Gordon L.: A stochastic approach to the gamma function. Amer. Math. Monthly 101 (1994), 858-865
19. Govedarica V. and Jovanović M.: Functions with quasiconvex derivatives. In: Recent Progress in Inequalities. Kluwer, Dordrecht 1998, 471-473
20. Hardy G. H., Littlewood J. E., and Pólya G.: Inequalities. Cambr. Univ. Press, Cambridge 1952
21. Ismail M. E. H., Lorch L., and Muldoon M. E.: Completely monotonic functions associated with the gamma function and its q-analogues. J. Math. Anal. Appl. 116 (1986), 1-9
22. Ismail M. E. H. and Muldoon M. E.: Inequalities and monotonicity properties for gamma and q-gamma functions. In: Approximation and Computation. Internat. Ser. Numer. Math. 119. Birkhäuser, Boston 1994, 309-323
23. Kahlig P. and Matkowski P.: Functional equations involving the logarithmic mean. Z. angew. Math. Mech. 76 (1996), 385-390
24. Kershaw D.: Some extensions of W. Gautschi's inequalities for the gamma function. Math. Comp. 41 (1983), 607-611
25. Leach E. B. and Sholander M. C.: Extended mean values. Amer. Math. Monthly 85 (1978), 84-90
26. Leach E. B. and Sholander M. C.: Extended mean values II. J. Math. Anal. Appl. 92 (1983), 207-223
27. Leach E. B. and Sholander M. C.: Multi-variable extended mean values. J. Math. Anal. Appl. 104 (1984), 390-407
28. Luke Y. L.: Inequalities for the gamma function and its logarithmic derivative. Math. Balkanica 2 (1972), 118-123
29. Merkle M.: Convexity, Schur-convexity and bounds for the gamma function involving the digamma function. Rocky Mount. J. Math. 28 (1998), 1053-1066
30. Mitrinović D. S.: Analytic Inequalities. Springer, New York 1970
31. Muldoon M. E.: Some monotonicity properties and characterizations of the gamma function. Aequationes Math. 18 (1978), 54-63
32. Niculescu C. P.: Convexity according to the geometric mean. Math. Inequal. Appl. 3 (2000), 155-167
33. Páles Zs.: Inequalities for differences of powers. J. Math. Anal. Appl. 131 (1988), 271-281
34. Páles Zs.: On two variable functional inequalities. C.R. Math. Rep. Acad. Sci. Canada 10 (1988), 25-28
35. Palumbo B.: Determinantal inequalities for the psi function. Math. Inequal. Appl. 2 (1999), 223-231
36. Pittenger A. O.: The logarithmic mean in n variables. Amer. Math. Monthly 92 (1985), 99-104
37. Roberts A. W. and Varberg D. E.: Convex Functions. Academic Press, New York 1973
38. Ronning G.: On the curvature of the trigamma function. J. Comp. Appl. Math. 15 (1986), 397-399
39. Sándor S.: On the gamma function III. Publ. C.R.M.P. Neuchâtel, Série 2, 19 (2001), 33-40
40. Spanier J. and Oldham K. B.: An Atlas of Functions. Hemisphere Publ. Co., Washington 1987
41. Stolarsky K. B.: Generalizations of the logarithmic mean. Math. Mag. 48 (1975), 87-92
42. Trimble S. Y., Wells J., and Wright F. T.: Superadditive functions and a statistical application. SIAM J. Math. Anal. 20 (1989), 1255-1259
43. Ubhaya V. A.: Quasi-convex optimization. J. Math. Anal. Appl. 116 (1986), 439-449