A Logarithmically Completely monotonic Function Involving the Gamma Functions 1

In this paper, sufficient conditions are found for a function involving the gamma function and its reciprocal to be logarithmically completely monotonic. Consequently, a decreasing monotonicity of the function is generalized and a known inequality is extended.

[1]  Christian Berg,et al.  Some classes of completely monotonic functions , 2002 .

[2]  Feng Qi (祁锋),et al.  A mononotonicity result of a function involving the gamma function , 2006 .

[3]  T. Teichmann,et al.  Harmonic Analysis and the Theory of Probability , 1957, The Mathematical Gazette.

[4]  Feng Qi (祁锋),et al.  Some Inequalities Involving the Geometric Mean of Natural Numbers and the Ratio of Gamma Functions , 2000 .

[5]  Feng Qi,et al.  Monotonicity of sequences involving geometric means of positive sequences with monotonicity and logarithmical convexity , 2006 .

[6]  Feng Qi,et al.  Monotonicity of sequences involving convex function and sequence , 2006 .

[7]  Christian Berg,et al.  Pick Functions Related to the Gamma Function , 2002 .

[8]  Feng Qi,et al.  Three classes of logarithmically completely monotonic functions involving gamma and psi functions , 2007 .

[9]  Monotonicity of Sequences Involving Geometric Means of Positive Sequences with Logarithmical Convexity , 2002 .

[10]  Feng Qi,et al.  Logarithmically completely monotonic functions relating to the gamma function , 2006 .

[11]  Eugene P. Wigner,et al.  Formulas and Theorems for the Special Functions of Mathematical Physics , 1966 .

[12]  H. Minc,et al.  Some Inequalities involving (r!)1/r , 1964, Proceedings of the Edinburgh Mathematical Society.

[13]  Chao-Ping Chen,et al.  Some completely monotonic functions involving the gamma and polygamma functions , 2006, Journal of the Australian Mathematical Society.

[14]  E. C. Titchmarsh,et al.  The Laplace Transform , 1991, Heat Transfer 1.

[15]  Feng Qi,et al.  INEQUALITIES AND MONOTONICITY FOR THE RATIO OF GAMMA FUNCTIONS , 2003 .

[16]  Feng Qi,et al.  A complete monotonicity property of the gamma function , 2004 .

[17]  Necdet Batir,et al.  Some Gamma Function Inequalities , 2006 .

[18]  Christian Berg,et al.  Some classes of completely monotonic functions, II , 2006 .

[19]  Christian Berg,et al.  Integral Representation of Some Functions Related to the Gamma Function , 2004 .

[20]  Christian Berg,et al.  A completely monotone function related to the Gamma function , 2001 .

[21]  Feng Qi (祁锋),et al.  TWO NEW PROOFS OF THE COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE PSI FUNCTION , 2009, 0902.2520.

[22]  R. Horn.  On infinitely divisible matrices, kernels, and functions , 1967 .

[23]  Feng Qi,et al.  Complete Monotonicities of Functions Involving the Gamma and Digamma Functions , 2004 .

[24]  K. Ball,et al.  Completely Monotonic Rational Functions and Hall's Marriage Theorem , 1994, J. Comb. Theory, Ser. B.

[25]  Horst Alzer,et al.  On some inequalities for the gamma and psi functions , 1997, Math. Comput..

[26]  C. Kimberling.  A probabilistic interpretation of complete monotonicity , 1974 .

[27]  Horst Alzer,et al.  Inequalities for the Gamma and Polygamma Functions , 1998 .

[28]  P. Davis.  Leonhard Euler's Integral: A Historical Profile of the Gamma Function: In Memoriam: Milton Abramowitz , 1959 .