An elementary proof of the hopf inequality for positive operators

Recently, OSTROWSKI [41 has established the connection between a theory of BIRXHOFF [1] on positive linear transformations and a theory of HOl'F [3] on positive linear integral operators. A central result of these investigations is an inequality of HOPF, which OSTROWSKI could extend to the more general case of positive linear transformations studied by BIRKHOFF. This inequality deals with the oscillation of a pair of positive vectors and gives a bound for the decrease of the oscillation under a positive linear transformation. I t leads to theoretical as well as practical applications of some importance especially in the theory of positive matrices E4], e.g. estimates for the modulus of the second eigenvalue in terms of the Perron root. A simplified derivation may therefore be worthwhile.