A generalized version of Holmstedt’s formula for the $K$-functional

Let $(A_0, A_1)$ be a compatible couple of quasi-normed spaces, and let $\Phi_0$ and $\Phi_1$ be two general parameters of the $K$-interpolation method. We compute the $K$-functional for the couple $((A_0,A_1)_{\Phi_0}, (A_0, A_1)_{\Phi _1})$ in terms of the $K$-functional for $(A_0, A_1)$.