Bounds on the joint and generalized spectral radius of Hadamard geometric mean of bounded sets of positive kernel operators

Let $\Psi _1, \ldots \Psi _m$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove that for the generalized spectral radius $\rho$ and the joint spectral radius $\hat{\rho}$ the inequalities $$\rho \left(\Psi _1 ^{\left( \frac{1}{m} \right)} \circ \cdots \circ \Psi _m ^{\left( \frac{1}{m} \right)} \right) \le \rho (\Psi _1 \Psi _2 \cdots \Psi _m) ^{\frac{1}{m}},$$ $$\hat{\rho} \left(\Psi _1 ^{\left( \frac{1}{m} \right)} \circ \cdots \circ \Psi _m ^{\left( \frac{1}{m} \right)} \right) \le \hat{\rho} (\Psi _1 \Psi _2 \cdots \Psi _m) ^{\frac{1}{m}}$$ hold, where $\Psi _1 ^{\left( \frac{1}{m} \right)} \circ \cdots \circ \Psi _m ^{\left( \frac{1}{m} \right)}$ denotes the Hadamard (Schur) geometric mean of the sets $\Psi _1, \ldots , \Psi _m$.