Monotonicity of a mean related to polygamma functions with an application

Let psi(n) = (-1)(n-1) psi((n)) (n = 0, 1, 2, ...), where psi((n)) denotes the psi and polygamma functions. We prove that for n >= 0 and two different real numbers a and b, the function X bar right arrow psi(-1)(n) (integral(b)(a) psi(n) (X + t)dt/b - a) - x is strictly increasing from (-min(a, b),infinity) onto (min(a, b), (a + b)/2), which generalizes a well-known result. As an application, the complete monotonicity for a ratio of gamma functions is improved.