An extension of an inequality for ratios of gamma functions

In this paper, we prove that for x+y>0 and y+1>0 the inequality [@C(x+y+1)/@C(y+1)]^1^/^x[@C(x+y+2)/@C(y+1)]^1^/^(^x^+^1^) 1 and reversed if x<1 and that the power 12 is the best possible, where @C(x) is the Euler gamma function. This extends the result of [Y. Yu, An inequality for ratios of gamma functions, J. Math. Anal. Appl. 352 (2) (2009) 967-970] and resolves an open problem posed in [B.-N. Guo, F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (2) (2003) 239-247].

[1]  Feng Qi (祁锋).  Bounds for the Ratio of Two Gamma Functions , 2009 .

[2]  Horst Alzer,et al.  Inequalities for the Volume of the Unit Ball in , II , 2000 .

[3]  M. J. Dubourdieu.  Sur un théorème de M. S. Bernstein relatif à la transformation de Laplace-Stieltjes , 1940 .

[4]  Feng Qi,et al.  On a new generalization of Martins' inequality , 2007 .

[5]  Inequalities and monotonicity of the ratio for the geometric means of a positive arithmetic sequence with arbitrary difference , 2003 .

[6]  Feng Qi (祁锋),et al.  Generalizations Of A Theorem Of I. Schur , 2006 .

[7]  Feng Qi,et al.  Wendel's and Gautschi's inequalities: Refinements, extensions, and a class of logarithmically completely monotonic functions , 2008, Appl. Math. Comput..

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

[9]  Martin E. Muldoon,et al.  Inequalities and monotonicity properties for gamma and q -gamma functions , 1994, 1301.1749.

[10]  Feng Qi (祁锋),et al.  PROPERTIES AND APPLICATIONS OF A FUNCTION INVOLVING EXPONENTIAL FUNCTIONS , 2009 .

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

[12]  C. Stegall.  Generalizations of a theorem of Namioka , 1988 .

[13]  M. E. Muldoon,et al.  Completely monotonic functions associated with the gamma function and its q-analogues , 1986 .

[14]  Horst Alzer,et al.  On an Inequality of H. Minc and L. Sathre , 1993 .

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

[16]  J. G. Wendel.  Note on the Gamma Function , 1948 .

[17]  Feng Qi (祁锋),et al.  Wendel-Gautschi-Kershaw's Inequalities and Sufficient and Necessary Conditions that a Class of Functions Involving Ratio of Gamma Functions are Logarithmically Completely Monotonic , 2007 .

[18]  Yaming Yu.  An inequality for ratios of gamma functions , 2009, 0909.3682.

[19]  J. Pečarić,et al.  On Van de Lune - Alzer's inequality , 2007 .

[20]  Binwu He,et al.  INEQUALITIES FOR THE VOLUME OF THE UNIT BALL IN ' n ⁄ , 2008 .

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

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

[23]  Feng Qi (祁锋).  BOUNDS FOR THE RATIO OF TWO GAMMA FUNCTIONS-FROM WENDEL'S AND RELATED INEQUALITIES TO LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS , 2009, 0904.1048.

[24]  H. Alzer.  On some inequalities involving (n!)1/n, II , 1994 .

[25]  Feng Qi,et al.  Monotonicity and logarithmic convexity relating to the volume of the unit ball , 2013, Optim. Lett..

[26]  Inequalities and monotonicity of the ratio of the geometric means of a positive arithmetic sequence with unit difference , 2003 .

[27]  Feng Qi,et al.  A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality , 2007 .

[28]  M. E. Muldoon.  Some monotonicity properties and characterizations of the gamma function , 1978 .

[29]  Feng Qi,et al.  A Logarithmically Completely monotonic Function Involving the Gamma Functions 1 , 2007 .

[30]  Li Wei,et al.  GENERALIZATIONS OF A THEOREM OF I. SCHUR , 2006 .

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

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

[33]  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 .

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

[35]  A COMPLETE SOLUTION TO AN OPEN PROBLEM RELATING TO AN INEQUALITY FOR RATIOS OF GAMMA FUNCTIONS , 2009 .

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