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We study analytic and geometric properties of Stieltjes and inverse Stieltjesfamilies defined on a separable Hilbert space and establish various minimalrepresentations for them by means of compressed resolvents of various types oflinear relations. Also attention is paid to some new peculiar properties ofStieltjes and inverse Stieltjes families, including an analog for the notion ofinner functions which will be characterized in an explicit manner. In addition,families which admit different types of scale invariance properties aredescribed. Two transformers that naturally appear in the Stieltjes and inverseStieltjes classes are introduced and their fixed points are identified.

Stieltjes and inverse Stieltjes holomorphic families of linear relations and their representations