The Furuta inequality is known as an exquisite extension of L\"owner-Heinz inequality. The grand Furuta inequality is its further extension including Ando-Hiai's inequality on log-majorization. We introduce two mean theoretic operator functions defined for positive invertible operators $X \leq A,$ for a given positive invertible operator $A$ and given $t \in [0,1]:$ $$\Phi(X) (=\Phi(u,p;X))=A^{-u} \sharp_{\frac{u+1}{u+p}} X^p(\le X)\ \ \ \text{for} \ \ \ u \ge 0,\ p \ge 1$$ and $$\Psi(X) (=\Psi(q,s;X))=(A^t \sharp_s X^q)^{\frac{1}{(1-s)t+sq}}(\le X)\ \ \ \text{for} \ \ \ q \ge 1,\ s \ge 1.$$ We have the grand Furuta inequality by using their composition and also have its further extension presented recently by Furuta, by using a successive composition of them.\\