On a functional equation related to two-variable Cauchy means

In this paper, we are dealing with the solution of the functional equation $$\phi \left(\frac{x+y}{2}\right)(f(x)-f(y))=F(x)-F(y),$$ concerning the unknown functions $\phi ,f$ and $F$ defined on a same open subinterval of the reals. Improving the previous results related to this topic, we describe the solution triplets $(\phi ,f,F)$ assuming only the continuity of $\phi$. As an application, under natural conditions, we also solve the equality problem of two-variable Cauchy means and two-variable quasi-arithmetic means.