Completely monotonic functions of positive order and asymptotic   expansions of the logarithm of Barnes double gamma function and Euler's gamma   function
  We introduce completely monotonic functions of order   $r>0$ and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order. 
Stamatis Koumandos and Henrik L. Pedersen