Transformation of positive functions by linear operators

In an earlier paper the author proved a conjecture of Vowels: iff/gis a positive function, so isf'/g', i.e., the positive property is preserved on differentiating the numerator and denominator. This paper generalizes this transformation process by the use of certain classes, of linear operators, both for general positive functions and for 2-element-type network functions, for which characterization theorems are first established.