Let \(F\) be an absolutely continuous distribution function with density \(f\) having support on the positive reals \(\mathbb{R}_+\). It is assumed that all the moments of \(F\) exist and are finite, but that they do not single out the distribution \(F\). The Stieltjes class for \(f\), see \textit{J. Stoyanov} [J. Appl. Probab. 41A, Spec. Issue, 281--294 (2004; Zbl 1070.60012)], is defined by \[ S(f,h):=\{f_{\varepsilon}= (1+\varepsilon\,h(x))\,f(x): \varepsilon\in\left[-1,1\right]\}, \] where \(h\) is defined on \(\mathbb{R}_+\) and is such that \(| h| \leq 1\). It is also assumed that the perturbation \(h\) satisfies \(\int_0^{+\infty} x^kf(x)h(x)\,dx=0\) for all \(k=0,1,\dots\). In the main theorem the function \(h\) is determined for the densities of the generalized gamma distribution defined by \( f(x)=K^{-1}x^{a-1}\exp\{-bx^c\}\), \(x>0, \) \(K\) being a normalizing constant. The index of dissimilarity \(D_S:=E[| h(X)| ]\) is calculated for Stieltjes classes of powers of the normal and of the exponential distributions.