Karlin's basic composition theorems and stochastic orderings

Suppose λ,x,ζ traverse the ordered sets λ,x and z,respectively and consider the functions f(λ,x,ζ) and g(λ,ζ) satisfying the following conditions,(a) f(λ,x,ζ) > 0 and f is tp2 in each pairs of variables when the third variable is held fixed; and (b) g(λ,ζ) is tp2. then the function [display equation],defined on λ × x is tp2 in (λ,x). the aim of this note is to use a new stochastic ordering argument to prove the above result and simplify it's proof given by karlin (1968). we also prove some other new versions of this result.