Concavity of weighted arithmetic means with applications

We prove that the following three conditions together imply the concavity of the sequence Pn iˆ0 aibi= Pn iˆ0 ai : concavity of fbng, log-concavity of fang and nonincreasing of f…bn ÿ bnÿ1†=…anÿ1=an ÿ anÿ2=anÿ1†g. As a consequence we get necessary and sufficient conditions for the concavity of the sequences fSnÿ1…x†=Sn…x†g and fSn…x†=Sn…x†g for any nonnegative x, where Sn…x† is the nth partial sum of a power series with arbitrary positive coefficients fang. 1. Introduction and results. Let sn…x† denote the nth partial sum of the Taylor series for the exponential function,