The monotonicity and convexity of a function involving psi function with applications

摘要

In this paper, we prove that the function x ↦ exp ( ψ ( x + 1 2 ) − 1 24 1 x 2 + 7 / 40 ) − x $$ xmapstoexp iggl( psi iggl( x+rac{1}{2} iggr) -rac {1}{24} rac{1}{x^{2}+7/40} iggr) -x $$ is decreasing from ( − 1 / 2 , ∞ ) $( -1/2,infty ) $ onto ( 0 , 1 / 2 ) $( 0,1/2 ) $ and convex on ( − 1 / 2 , ∞ ) $( -1/2,infty ) $ . As a consequence of the main theorem, various type of bounds for the psi function are presented, which essentially generalize or improve some known results.