Some scales of equivalent conditions to characterize the Stieltjes inequality: The case q < p

We prove that the weighted Stieltjes inequality, can be characterized by four different scales of conditions also for the case 0 < q < p < ∞, 1<p. In particular, a new proof of a result of G. Sinnamon  is given, which also covers the case 0<q<1. Moreover, for the situation at hand a new gluing theorem and also an equivalence theorem of independent interest are proved and discussed. By applying our new equivalence theorem to weighted inequalities for the Stieltjes transform we obtain the four new scales of conditions for characterization of Stieltjes inequality. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.