= Monotonicity and bounds of effective bandwidth
{title2=$\mathbb EX\le\log\mathbb Ee^{sX}/s\le\operatorname{ess\,sup}X$}
For $s>0$ in the finite <moment-generating function> domain and integrable $X$, <Jensen inequality> gives the lower bound and $X\le\operatorname{ess\,sup}X$ almost surely gives the upper bound. The <Holder inequality> makes $K(s)=\log\mathbb Ee^{sX}$ convex with $K(0)=0$. Thus $K(t)\le(t/s)K(s)$ for $0<t<s$, and the <effective bandwidth> is nondecreasing. It need not be strictly increasing: deterministic traffic gives a constant bandwidth. Infinite peaks or exponential moments require extended-value conventions.
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