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This paper shows the strong converse and the dispersion of memorylesschannels with cost constraints and performs refined analysis of the third orderterm in the asymptotic expansion of the maximum achievable channel coding rate,showing that it is equal to $\frac 1 2 \frac {\log n}{n}$ in most cases ofinterest. The analysis is based on a non-asymptotic converse bound expressed interms of the distribution of a random variable termed the $\mathsf b$-tiltedinformation density, which plays a role similar to that of the $\mathsfd$-tilted information in lossy source coding. We also analyze the fundamentallimits of lossy joint-source-channel coding over channels with costconstraints.

Channels with cost constraints: strong converse and dispersion