The polygamma functions are derivatives of the logarithmic derivative \(\psi\) of the gamma function. As one of the main results necessary and sufficient conditions are offered for the inequality \(|\psi^{(k)} [M^r (\mathbf{x},\mathbf{p})]|\leq M^s (|\psi^{(k)}(\mathbf{x)}|,\mathbf{p})\) to hold for all \(\mathbf{x}=(x_1,\dots ,x_n)\in ]0,\infty[^n\), \(\mathbf{p}=(p_1,\dots ,p_n)\in\{\mathbf{p}\in]0,\infty[^n\mid p_1 +\dots +p_n =1\}\) \(k\geq 1, n\geq 2\) fixed integers), where \(M^t(\mathbf{x},\mathbf{p})\) is the \(t\)-th power mean of \(x_1,\dots ,x_n\) with weights \(p_1,\dots ,p_n\) (including the weighted geometric mean if \(t=0\)). If these conditions connecting \(r\) and \(s\) are satisfied then there is equality in the above inequality iff \(x_1=\dots =x_n.\)