Some Diophantine equations involving arithmetic functions and Bhargava factorials

F. Luca proved for any fixed rational number $\alpha>0$ that 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 the mentioned 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 do so by including the case $f$ is the sum of $k$\textsuperscript{th} powers of divisors function. Moreover, we observe that the same holds by replacing some of the factorials with certain examples of Bhargava factorials.