In the paper, by virtue of two new tools, the authors present the monotonicity and convexity of certain combinations of the complete elliptic integrals of the first kind, and obtain new sharp bounds and inequalities for them.

In this paper, by using the monotonicity rule for the ratio of two Laplace transforms, we prove that the function x ↦ 1 24 x ( ln Γ ( x + 1 / 2 ) − x ln x + x − ln 2 π ) + 1 − 120 7 x 2 $$ x\mapsto \frac{1}{24x ( \ln \Gamma ( x+1/2 ) -x\ln x+x- \ln \sqrt{2\pi } ) +1}-\frac{120}{7}x^{2} $$ is strictly increasing from ( 0 , ∞ ) $( 0,\infty ) $ onto ( 1 , 1860 / 343 ) $( 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.