Let $(M^n,g)$ be complete, let $p\in M$, and let $r=r_p$. Let $\Omega_p$ denote the regular domain of $p$, namely the complement of $p$ and its cut locus, and let $x\in \Omega_p$ with $r(x)=t$. Assume $\operatorname{sn}_k(s)>0$ for every $s\in(0,t]$ (for $k>0$, this is equivalent to $t<\pi/\sqrt{k}$). Here $\operatorname{sn}_k$ is the solution of $\operatorname{sn}_k''+k\operatorname{sn}_k=0$, $\operatorname{sn}_k(0)=0$, $\operatorname{sn}_k'(0)=1$, and $\operatorname{ct}_k=\operatorname{sn}_k'/\operatorname{sn}_k$ on the interval where $\operatorname{sn}_k>0$. If every radial sectional curvature along the minimizing geodesic from $p$ to $x$ satisfies $K(\dot\gamma,E)\ge k$, then for every $X\in T_xM$ with $g(X,\nabla r)=0$,

\begin{align*} \operatorname{Hess}r(X,X)\le \operatorname{ct}_k(t)|X|^2. \end{align*}

If every radial sectional curvature satisfies $K(\dot\gamma,E)\le k$, then

\begin{align*} \operatorname{Hess}r(X,X)\ge \operatorname{ct}_k(t)|X|^2. \end{align*}