Monotonicity, convexity, and complete monotonicity of two functions related to the gamma function

In this paper, we prove that for a = 31/ 98, the function fa (x) = ln x + 1 2 -x ln x + x -1 2 ln (2p) + x 24 x2 + a -7/120 x4 + ax2 + (98a -31) /1680 is strictly increasing (decreasing) and concave (convex) on (0,8) if and only if a = 5281/ 6068 (a = 31/ 98). Moreover, we show that the necessary and sufficient condition for function Fa (x) = - x4 + ax2 + 98a -31 1680 fa (x) for a. R to be completely monotonic on ( 0,8) is also a = 5281/ 6068. These yield some new sharp bounds for the gamma function.