Improved interpolation theorems for a class of linear operators

New interpolation theorems are formulated for the class of operators that map the cone of positive functions into the cone of monotonic decreasing functions. These theorems are based on the concept of the $K^c$-functional. A formula for calculating the $K^c$-functional for some pairs of spaces is suggested. An example of an interpolation pair of spaces is considered in which the cones obtained with the help of Peetre's $K$-functional are different from those obtained using the $K^c$-functional.