By means of the concepts of the same order and the oppoite order between functions, studiedthe properties of weighted powr means of functions and their functional characteristics such as the loga-rithmic convexity and monotony when they are considered as functions of their power exponents. As appli-cations a new mean was introduced, comparisons between the so-called Holder's means and the Sto-larsky's means were made, and results of the refinements of the Ostle-Tewlliger inequality and of thearithmtic-geometric mean inequality were obtained.