Let \(\psi_{n}= ( -1 ) ^{n-1}\)\(\psi^{ ( n ) }\) (\(n=0,1,2,\ldots \)), where \(\psi^{ ( n ) }\) denotes the psi and polygamma functions. We prove that for \(n\geq0\) and two different real numbers a and b, the function $$ x\mapsto\psi_{n}^{-1} \biggl( \frac{\int_{a}^{b}\psi_{n}(x+t)\,dt}{b-a} \biggr) -x $$ is strictly increasing from \(( -\min ( a,b ) ,\infty ) \) 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.