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Recently, several authors have proved inequalities on the spectral radius$\rho$, operator norm $\|\cdot\|$ and numerical radius of Hadamard products andordinary products of non-negative matrices that define operators on sequencespaces, or of Hadamard geometric mean and ordinary products of positive kerneloperators on Banach function spaces. In the present article we generalize andrefine several of these results. In particular, we show that for a Hadamardgeometric mean $A ^{\left( \frac{1}{2} \right)} \circ B ^{\left( \frac{1}{2}\right)}$ of positive kernel operators $A$ and $B$ on a Banach function space$L$, we have $$ \rho \left(A^{(\frac{1}{2})} \circ B^{(\frac{1}{2})} \right)\le \rho \left((AB)^{(\frac{1}{2})} \circ(BA)^{(\frac{1}{2})}\right)^{\frac{1}{2}} \le \rho (AB)^{\frac{1}{2}}.$$ In thespecial case $L=L^2(X, \mu)$ we also prove that $$\|A^{(\frac{1}{2})} \circB^{(\frac{1}{2})} \| \le \rho \left ( (A^* B ) ^{(\frac{1}{2})}\circ (B ^*A)^{(\frac{1}{2})}\right)^{\frac{1}{2}} \le \rho (A^* B )^{\frac{1}{2}}.$$

Inequalities on the spectral radius, operator norm and numerical radius of Hadamard weighted geometric mean of positive kernel operators