Monotonicity and inequalities for the gamma function

In this paper, by using the monotonicity rule for the ratio of two Laplace transforms, we prove that the function $$ x\mapsto \frac{1}{24x ( \ln \Gamma ( x+1/2 ) -x\ln x+x- \ln \sqrt{2\pi } ) +1}-\frac{120}{7}x^{2} $$ is strictly increasing from $( 0,\infty ) $ onto $( 1,1860/343 ) $ . This not only yields some known and new inequalities for the gamma function, but also gives some completely monotonic functions related to the gamma function.