Monotonicity properties of a function involving the psi function with applications

In this paper, we present the best possible parameter a is an element of (1/5, infinity) such that the functions psi' (x + 1) - L-x(x,a) and psi ''(x + 1) - L-xx(x,a) are strictly increasing or decreasing with respect to x is an element of(0,infinity), where L(x,a) = 1/90a(2)+2 log(x(2) + x + 3a+1/3) + 45a(2)/90a(2)+2 log(x(2) + x + 15a-1/45a) and psi(x) is the classical psi function. As applications, we get several new sharp bounds for the psi function and its derivatives.