Is the geometric average of two densities functions a density function?

Question

Following the question of the topic Multiplication of two density functions, i wonder if the weighted geometric average of two probability density functions (pdfs) is itself a density ? More formally, if i have two pdfs $f_1(x)$ and $f_2(x)$, is the product function $g(x)$ defined like: $$g(x) = f_1(x)^{a} \times f_2(x)^{1-a}$$ a density with $a$ a real number not necessearily comprised between $0$ and $1$ ?