Logarithmically completely monotonic functions concerning gamma and digamma functions

Abstract

For given real numbers a0, b and c, let Fa, b, c(x)=[(x+1)]1/x(1+a/x)x+b/xc and a, b, c(x)=''(x)+[2+(b+c)x-2x2]/x3+[3a(2a-b)+(6a-b)x+2x2]/(x+a)3 with x(0, ), where (x) and (x) are the well-known Euler gamma function and the psi or digamma function, respectively. In this article, it is revealed that the function Fa, b, c(x) for 2a3b-3c and its reciprocal 1/Fa, b, c(x) for 2a3b and 1+2b+c0 are logarithmically completely monotonic in (0, ), while the function a, b, c(x) for 02a3b and 1+2b+c0 and its negative-a, b, c(x) for 02a3b and b+c0 are completely monotonic in (0, ).