New means of Cauchy's type

We will introduce new means of Cauchy's type M(r,l)(s) (f, mu) defined, for example, as M(r,l)(s) (f, mu) = ((l(l-s)/r(r-s))(M(r)(r)(f, mu) - M(s)(r)(f, mu)/M(l)(l)(f, mu)-M(s)(l)(f, mu)))(1/(r-l)),in the case when l not equal r not equal s, l, r not equal 0. We will show that this new Cauchy's mean is monotonic, that is, the following result. Theorem. Let t, r, u, v is an element of R, such that t <= v, r <= u. Then forM(r,l)(s) (f, mu), one has M(t,r)(s) <= M(v,u)(s). We will also give some related comparison results. Copyright (C) 2008 M. Anwar.