How can I check whether the solution $y=1$ to the integral equation $1=\int_a^b y^{1 / x} f(x) \ \mathrm{d}x\,, $ is unique?

Question

I'm trying to solve the following equation for $y$: $$ 1 = \int_a^b y^{1 / x} f(x) \ \mathrm{d}x\,, $$ where $f(x)$ is a density function with support over the interval $[a,b]$, so that $\int_a^b f(x) \ \mathrm{d}x = 1$. Clearly, $y=1$ is a solution, but how can I check whether the solution $y=1$ is unique?

For context, in my application I assume that $f(x)$ is the density function for the Pareto distribution.