In optical wireless communications, a broadcast channel (BC) employing intensity modulation and direct detection (IM/DD) is often modelled as a peak-constrained BC. A closed-form expression for its capacity region of the peak-constrained BC is not known. This paper presents an analytical capacity inner bound for the peak-constrained Gaussian BC achieved by a class of discrete input distribution, specifically, the evenly-spaced discrete uniform distribution (ESDU). In contrast to the continuous input distribution that provides the benchmark, ESDU is more promising in the application of peak-constrained Gaussian channels. The newly obtained capacity inner bound is easily-computable and is numerically shown to be tighter than the benchmark. Besides, we remark the newly developed analytical upper bound for the ESDU rate, which is tight in all tested settings.