The author generalizes Stolarsky's Extended Mean Values to a four-parameter family of means F(r,s;a,b;x,y) = E(r,s;ax,by)/E(r,s;a,b) and investigates their monotonicity properties.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm100-1-11, author = {Alfred Witkowski}, title = {Weighted extended mean values}, journal = {Colloquium Mathematicae}, volume = {100}, year = {2004}, pages = {111-117}, zbl = {1051.26022}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm100-1-11} }
Alfred Witkowski. Weighted extended mean values. Colloquium Mathematicae, Tome 100 (2004) pp. 111-117. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm100-1-11/