CONCAVITY OF WEIGHTED ARITHMETIC MEANS WITH APPLICATIONS

We prove that the following three conditions together imply the concav ity of the sequence {(i=0)Sigma(n) alpha(i) beta(i)/(i=0)Sigma(n) alph a(i)}: concavity of {beta(n)}, log-conavity of {alpha(n)} and nonincre asing of -1))/(alpha(n-1)/alpha(n)-alpha(n-2)/alpha(n-1))}. As a conse quence we get necessary and sufficient conditions for the concavity of the sequences {Sn-1(x)/S-n(x)} and {S'(n)(x)/S-n(x)} for any nonnegat ive x, where S-n(x) is the nth partial sum of a power series with arbi trary positive coefficients {alpha(n)}.