Monotonicity and inequalities for the gamma function

Abstract

Abstract 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.

Keywords

• gamma function
• Laplace transform
• complete monotonicity
• inequality