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Set $$\theta (s/t): = (s/t - 1)(t/s)^{\frac{{s/t}}{{s/t - 1}}} = (s - t)\frac{{t^{t/(s - t)} }}{{s^{s/(s - t)} }}$$ if 0<$t$<$s$. The key result of the paper shows that if ($T (t))$_{$t$>0}is a nontrivial strongly continuous quasinilpotent semigroup of bounded operators on a Banach space then there exists δ>0 such that ║$T(t)-T(s)$║>θ(s/t) for 0<$t$<$s$≤δ. Also if ($T(t)$)_{$t$>0}is a strongly continuous semigroup of bounded operators on a Banach space, and if there exists η>0 and a continuous function$t$→$s(t)$on [0, ν], satisfying$s$(0)=0, and such that 0<$t$<$s(t)$and ║$T(t)-T(s(t))║<θ(s/t)$for$t$∈(o, η], then the infinitesimal generator of the semigroup is bounded. Various examples show that these results are sharp.

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