We find that Regge trajectories cannot increase faster than linearly as $s\to \infty$ in any direction on the physical sheet of the complex $s$ plane. Further, if $\mathrm{Re}\alpha \mathrm{}\left(s\right)\mathrm{}$ increases more rapidly than $\mathrm{Im}\alpha \mathrm{}\left(s\right)\mathrm{}$ as $s\to +\infty$, the trajectories must be asymptotically linear. These results are valid only to within possible logarithmic factors and are a consequence of the generally assumed absence of resonance poles on the physical sheet. Other assumptions are as follows. (i) $\alpha \mathrm{}\left(s\right)\mathrm{}$ is real analytic in the $s$ plane cut from threshold ($s_0$) to $+\infty$. (ii) $\mathrm{Im}\alpha \mathrm{}\left(s\right)\mathrm{}\ge 0$ for $s_0\le s<+\infty$. (iii) $\alpha \mathrm{}\left(s\right)\mathrm{}$ is continuous on the cut and increases no faster than a power of $s$ as $s\to +\infty$, i.e., along the positive real axis. (iv) As $s\to \infty$ in complex directions, $\left|\alpha \right|$ increases no faster than a power of $\exp \mathrm{}\left(\right|s\left|\right)\mathrm{}$.

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