On the directional growth of the resolvent norm

Let $A$ be a closed densely defined operator on a separable Hilbert space $\mathcal{H}$. Assume the resolvent set $\rho(A)$ is non-empty. For $z,z'\in\rho(A)$ let $[z,z']$ denote the straight line segment from $z$ to $z'$. For each $z\in\rho(A)$ we classify the behavior of the resolvent norm $\zeta\mapsto\lVert R_A(\zeta) \rVert$ near $z$. Either there are $z'\in\rho(A)$, $z'\neq z$, $[z,z']\subset\rho(A)$, such that $\lVert R_A(\zeta) \rVert \geq \lVert R_A(z) \rVert + C\lvert \zeta-z \rvert^\delta$ for $\zeta\in[z,z']$ with $\delta=1$ or $\delta=2$, or the function $\zeta\mapsto\lVert R_A(\zeta) \rVert$ has a global minimum at $\zeta=z$.