Monotonicity and inequalities for the gamma function

Journal of Inequalities and Applications

Jing-Feng Tian¹  Zhen-Hang Yang¹ 

[1] College of Science and Technology, North China Electric Power University;

关键词: gamma function;    Laplace transform;    complete monotonicity;    inequality;   

DOI  :  10.1186/s13660-017-1591-9

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown