= Solution
For the distinct observed event times $a_k$, let $i_k=\pi(a_k)$ and let $R_k$ be the <risk set> immediately before that event. The <Cox partial likelihood> is
$$
L_p(\beta)=\prod_{k=1}^d\frac{e^{\beta z_{i_k}}}{\sum_{i\in R_k}e^{\beta z_i}},\qquad
\ell_p(\beta)=\sum_{k=1}^d\left\{\beta z_{i_k}-\log S_0(a_k,\beta)\right\}.
$$
The different risk sets already incorporate each event and any intervening censoring. Differentiating this <log-likelihood> gives
$$
U(\beta)=\ell_p'(\beta)=\sum_{k=1}^d\left[z_{i_k}-\frac{S_1(a_k,\beta)}{S_0(a_k,\beta)}\right]
=\sum_{k=1}^d s(a_k,\beta).
$$
Therefore a finite interior <maximum-likelihood estimate> from the <Cox partial likelihood> satisfies the first-order condition
$$
\boxed{\sum_{k=1}^d s(a_k,\widehat\beta)=0.}
$$
Moreover
$$
-\ell_p''(\beta)=\sum_{k=1}^d\left[\frac{S_2(a_k,\beta)}{S_0(a_k,\beta)}-\left(\frac{S_1(a_k,\beta)}{S_0(a_k,\beta)}\right)^2\right]\geq0.
$$
Each summand is the conditional event-covariate <variance>, so the <partial likelihood> is log-concave. If some informative risk set has unequal covariates, the curvature is strictly negative and an existing finite root is the unique maximizer.
The finite-interior qualification is necessary. With two at-risk subjects having covariates zero and one, suppose the subject with covariate one is the sole observed event and the other is subsequently censored. Then $L_p(\beta)=e^\beta/(1+e^\beta)$ increases strictly and has its supremum only as $\beta\to+\infty$; the score $1/(1+e^\beta)$ has no finite zero. If every event risk set has identical covariates, the coefficient is instead unidentifiable and the score is identically zero. The displayed fitted-score equation applies to the regular case in which the proportional-hazards estimate is finite.
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