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Let $\Psi$ and $\Sigma$ be bounded sets of positive kernel operators on aBanach function space $L$. We prove several refinements of the knowninequalities $$\rho \left(\Psi ^{\left( \frac{1}{2} \right)} \circ \Sigma^{\left( \frac{1}{2} \right)} \right) \le \rho (\Psi \Sigma) ^{\frac{1}{2}}\;\; \mathrm{and}\;\; \hat{\rho} \left(\Psi ^{\left( \frac{1}{2} \right)} \circ\Sigma ^{\left( \frac{1}{2} \right)} \right) \le \hat{\rho} (\Psi \Sigma)^{\frac{1}{2}} $$ for the generalized spectral radius $\rho$ and the jointspectral radius $\hat{\rho}$, where $\Psi ^{\left( \frac{1}{2} \right)} \circ\Sigma ^{\left( \frac{1}{2} \right)} $ denotes the  Hadamard (Schur) geometric mean of the sets $\Psi $ and $\Sigma$.Furthermore, we prove that analogous inequalities hold also for the generalizedessential spectral radius and the joint essential spectral radius in the casewhen $L$ and its Banach dual $L^*$ have order continuous norms.

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