Inequality for second characteristic values of positive operators of certain classes

A homogeneous additive operator $A$, positive on a cone $K$ of a Banach space $E$ partially ordered by $K$, is investigated. It is assumed that $K$ is a reproducing cone in $E$ and that $A$ has a characteristic vector $u_0: Au_0=\lambda_0u_0$ in $K$. It is proved that if $AK\subset K_{u_0,\rho}$ for some $\rho\geqslant1$, then any other characteristic value $\lambda$ of $A$ satisfies the inequality $|\lambda|<(\rho-1)/(\rho+1)\lambda_0$. This is the best possible upper bound in the class of operators considered.