 Solution

Within the risk set $R_i$, the Cox proportional-hazards model makes the conditional failure probabilities proportional to $e^{\beta z_j}$. ThusThe score function sums observed minus expected covariates at event times. At a finite interior maximum of the Cox partial likelihood,The balance holds in aggregate, not separately at every event. The log partial likelihood is concave becauseWhen there is informative covariate variation in the risk sets, this gives a unique finite solution if it exists. A completely separated event pattern can instead have its maximum only as $\beta \to \pm \infty$; the question's finite $\hat{\beta }$ implicitly excludes that case.