The function 1 divided by "x"[superscript 2] minus "e"[superscript"-x"] divided by (1 minus "e"[superscript"-x"])[superscript 2] for "x" greater than 0 is proved to be strictly decreasing. As an application of this monotonicity, the logarithmic concavity of the function "t" divided by "e"[superscript "at"] minus "e"[superscript"(a-1)""t"] for "a" as an element of the set of real numbers and "t" as an element of (0,infinity) is verified. The possible origin and background of the function (*) are revealed to be related to theta("x") = integral[superscript infinity;subscript 0](1 divided by "e"[superscript "t"] minus 1, minus 1 divided by "t" plus 1 divided by 2)"e"[superscript "-xt"] divided by "t" times "dt," the remainder of Binet's formula. Some applications of above results to the difference of theta(x) are noted.