  In the article, a notion "logarithmically absolutely monotonic function" is
introduced, an inclusion that a logarithmically absolutely monotonic function
is also absolutely monotonic is revealed, the logarithmically complete
monotonicity and the logarithmically absolute monotonicity of the function
$\bigl(1+\frac{\alpha}x\bigr) ^{x+\beta}$ are proved, where $\alpha$ and
$\beta$ are given real parameters, a new proof for the inclusion that a
logarithmically completely monotonic function is also completely monotonic is
given, and an open problem is posed.
