A Complete Monotonicity of the Gamma Function

The function 1/x ln Γ(x+1)−ln x+1 is strictly completely monotonic on (0,∞)

Logarithmically Completely Monotonic Ratios of Mean Values and an Application

In the article, some strictly Logarithmically completely monotonic ratios of mean values are presented

Monotonicity and Inequalities for Ratio of the Generalized Logarithmic Means

Let c > b > a > 0 be real numbers. Then the function f(r) = Lr(a,b)/Lr(a,c) is strictly decreasing on (−∞,∞), where Lr(a, b) denotes the generalized (extended) logarithmic mean of two positive numbers a and b

Constraints on the Brans-Dicke gravity theory with the Planck data

Based on the new cosmic CMB temperature data from the Planck satellite, the 9 year polarization data from the WMAP, the BAO distance ratio data from the SDSS and 6dF surveys, we place a new constraint on the Brans-Dicke theory. We adopt a parametrization \zeta=\ln(1+1/\omega}), where the general relativity (GR) limit corresponds to $\zeta = 0$. We find no evidence of deviation from general relativity. At 95% probability, $-0.00246 < \zeta < 0.00567$, correspondingly, the region $-407.0 < \omega <175.87$ is excluded. If we restrict ourselves to the $\zeta>0$ (i.e. $\omega >0$) case, then the 95% probability interval is $\zeta 181.65$. We can also translate this result to a constraint on the variation of gravitational constant, and find the variation rate today as $\dot{G}=-1.42^{+2.48}_{-2.27} \times 10^{-13}$yr$^{-1}$ ($1\sigma$ error bar), the integrated change since the epoch of recombination is $\delta G/G = 0.0104^{+0.0186}_{-0.0067}$ ($1\sigma$ error bar). These limits on the variation of gravitational constant are comparable with the precision of solar system experiments.Comment: 7 pages, 5 figures, 2 table