Optimal Lehmer Mean Bounds for the Combinations of Identric and Logarithmic Means

摘要

For anyα∈0,1, weanswer the questions: what are the greatest valuespandλand the least valuesqandμ, such that theinequalitiesLpa,b<Iαa,bL1-αa,b<Lqa,bandLλa,b<αIa,b+1-αLa,b<Lμa,bholdfor alla,b>0witha≠b? Here,Ia,b,La,b, andLpa,bdenote the identric, logarithmic, andpth Lehmermeans of two positive numbersaandb, respectively.