Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic

The psi function @j(x) is defined by @j(x)[email protected]^'(x)/@C(x), where @C(x) is the gamma function. We give necessary and sufficient conditions for the function @j^''(x)+[@j^'([email protected])]^2 or its negative to be completely monotonic on ([email protected],~), where @[email protected]?R. We also prove that the function [@j^'(x)]^[email protected]@j^''(x) is completely monotonic on (0,~) if and only if @l= 0 and [email protected]?R.