  The next-to-leading order cross section for the inclusive production of charm
quarks in gamma-gamma collisions is calculated as a function of the transverse
momentum pT and the rapidity y in approaches using massive or massless charm
quarks. For the direct cross section we derive the massless limit from the
massive theory with the result that this limit differs from the massless
version with MSbar factorization by finite corrections. Subtracting or adding
these corrections allows us to compare the two approaches on equal footing. We
establish massless and massive versions with 3 and 4 initial flavours which are
shown to approach the massless approximations very fast with increasing pT.
With these results we calculate the inclusive D* cross section in gamma-gamma
collisions using realistic evolved fragmentation functions with appropriate
factorization scales and compare with recent data for dsigma/dpT from three LEP
collaborations after single- and double-resolved contributions have been added.
