Relativistic plasma emission due to a beam instability in the polar cap regions is examined critically as a pulsar radio emission mechanism. Wave dispersion in the pulsar plasma is discussed, based on the use of a relativistic plasma dispersion function. The growth rate for the beam instability is estimated in the rest frame of the plasma for parallel Langmuir waves, L-O mode waves, and oblique Alfvén waves. The first two of these imply frequencies that are much higher than the observed frequencies for plausible parameters, suggesting that they are not viable as pulsar radio emission mechanisms. Growth of Alfvén waves requires that the beam speed equal the phase speed of the Alfvén waves, and this condition cannot be satisfied within the light cylinder, except for an extremely high energy beam. It is suggested that either the plasma parameters in the source region are quite different from what is currently considered plausible or the emission mechanism does not involve a beam instability. Alternative pulsar radio emission mechanisms should be explored further.