  In this paper we shall consider some famous means such as arithmetic,
harmonic, geometric, root square mean, etc. Considering the difference of these
means, we can establish. some inequalities among them. Interestingly, the
difference of mean considered is convex functions. Applying some properties,
upper bounds on the probability of error are established in this paper. It is
also shown that the results obtained are sharper than obtained directly
applying known inequalities.
