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We consider differences between $\log \Gamma(x)$ and truncations of certainclassical asymptotic expansions in inverse powers of $x-\lambda$ whosecoefficients are expressed in terms of Bernoulli polynomials $B_n(\lambda)$,and we obtain conditions under which these differences are strictly completelymonotonic. In the symmetric cases $\lambda=0$ and $\lambda=1/2$, we recoverresults of Sonin, N\"orlund and Alzer. Also we show how to derive theseasymptotic expansions using the functional equation of the logarithmicderivative of the Euler gamma function, the representation of $1/x$ as adifference $F(x+1)-F(x)$, and a backward induction.

Bounds for the logarithm of the Euler gamma function and its derivatives