  This paper is motivated by an astonishing result of H. Alzer and S.
Ruscheweyh published in 2001 in the Proc. Amer. Math. Soc., which states that
the intersection of the classes two-variable Gini means and Stolarsky means is
equal to the class of two-variable power means. The two-variable Gini and
Stolarsky means form two-parameter classes of means expressed in terms of power
functions. They can naturally be generalized in terms of the so-called
Bajraktarevi\'c and Cauchy means. Our aim is to show that the intersection of
these two classes of functional means, under high-order differentiability
assumptions, is equal to the class of two-variable quasiarithmetic means.
