Some Diophantine equations involving arithmetic functions and Bhargava factorials

F. Luca proved that for any fixed rational number $\alpha \gt 0$ the Diophantine equations of the form $\alpha m!=f(n!)$, where $f$ is either the Euler function or the divisor sum function or the function counting the number of divisors, have only finitely many integer solutions $(m,n)$. In this paper we generalize that result and show that Diophantine equations of the form $\alpha m_1!\cdots m_r!=f(n!)$ have finitely many integer solutions, too. In addition, we include the case where $f$ is the sum-of-$k$th-powers-of-divisors function. Moreover, the same holds on replacing some of the factorials with certain Bhargava factorials.