Complete Monotonicities of Functions Involving the Gamma and Digamma Functions

In the article, the completely monotonic results of the functions [Γ(x+ 1)]1/x, [Γ(x+α+1)]1/(x+α) [Γ(x+1)]1/x , [Γ(x+1)]1/x (x+1)α and [Γ(x+1)]1/x xα in x ∈ (−1,∞) for α ∈ R are obtained. In the final, three open problems are posed.

[1]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

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

[3]  Feng Qi,et al.  Monotonicity and Convexity Results for Functions Involving the Gamma Function , 2003 .

[4]  Feng Qi,et al.  The Extended Mean Values: Definition, Properties, Monotonicities, Comparison, Convexities, Generalizations, and Applications , 2001 .

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

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

[7]  J. Voigt,et al.  A monotonicity property of the -function. , 2002 .

[8]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[9]  Feng Qi,et al.  The function $(b^x-a^x)/x$: Inequalities and properties , 1998 .

[10]  K. S. Kölbig,et al.  Errata: Milton Abramowitz and Irene A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U.S. Government Printing Office, Washington, D.C., 1994, and all known reprints , 1972 .

[11]  D. Widder,et al.  The Laplace Transform , 1943, The Mathematical Gazette.

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

[13]  Feng Qi (祁锋),et al.  A Complete Monotonicity of the Gamma Function , 2004 .

[14]  Peter S. Bullen,et al.  A Dictionary of Inequalities , 1998 .

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

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

[17]  Feng Qi,et al.  Logarithmic convexity of extended mean values , 2001 .

[18]  D. S. Mitrinovic,et al.  Classical and New Inequalities in Analysis , 1992 .

[19]  Feng Qi,et al.  Monotonicity Results and Inequalities for the Gamma and Incomplete Gamma Functions , 2002 .

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

[21]  Feng Qi,et al.  Monotonicity Results For The Gamma Function , 2002 .

[22]  MONOTONICITY AND CONVEXITY OF THE FUNCTION , 2004 .