Let $R$ be a root system (irreducible if that makes this easier) in the real vectorspace $E$.
Let $\lambda$ and $\mu$ in $E$ with $w_0(\lambda)\leq \mu \leq \lambda$ where $w_0$ is the longest element in the Weyl group of $R$ and the ordering is the usual one coming from $R$. Assume that $\langle\lambda,\alpha^{\vee}\rangle\geq 0$ for all simple roots $\alpha$.
Let $\alpha_0$ be the highest short root of $R$. Is it then true that $\langle\mu,\alpha^{\vee}\rangle \geq -\langle\lambda,\alpha_0^{\vee}\rangle$ for any simple root $\alpha$? Or is there some other similar bound that holds?
Motivation: In my setup, $\mu$ is a weight of the simple module $L(\lambda)$ for a simple, simply connected algebraic group, and I would like to find a weight $\nu$ which is large enough such that $\langle\nu+\mu,\alpha^{\vee}\rangle\geq -1$ for all simple roots $\alpha$, or more precisely, figure out how large I need to make $\nu$ in terms of $\lambda$ for this to hold for all weights $\mu$ of $L(\lambda)$.