In the paper, the authors concisely survey and review some functions involving the gamma function and its various ratios, simply state their logarithmically complete monotonicity and related results, and find necessary and sufficient conditions for a new function involving the ratio of two gamma functions and originating from the coding gain to be logarithmically completely monotonic.
We shall show that, for any given point p on a Riemannian manifold (M, g), there is a pointwise conformai metric g = g in which thê-geodesic sphere centered at p with radius r has constant mean curvature 1/r for all sufficiently small r. Furthermore, the exponential map of g at p is a measure preserving map in a small ball around p .
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