... CorporationJournal of Inequalities and ApplicationsVolume 2007, Article ID 37359, 8 pagesdoi:10.1155/2007/37359 Research Article On Logarithmic Convexity for Differences of Power MeansSlavko SimicReceived ... andpigxi≥gpixi(4.3) for each xi∈ I, i = 1,2, , and any positive weight sequence {pi},pi= 1.Lemma 4.4. If φ(s) is continuous and convex for all s1, s2, s3of an open interval I for whichs1<s2<s3, ... nonnegative and integrable functions for x∈ (a,b),withbap(x)dx = 1.DenoteDs= Ds(a,b, f , p):=bap(x) fs(x) dx −bap(x) f (x)dxs. (2.4) For 0 <r<s<t, r,s,t=1,...