We present new spectroscopic observations of Mkn 309, a starburst galaxy with one of the largest WR populations known. A highly super solar metallicity is derived. Using additional objects we analyse a sample of five metal-rich WR galaxies with the main goal of constraining the basic properties of the massive star populations (IMF slope, M_up) and the star formation history (age, burst duration) of these objects by quantitative comparisons with evolutionary synthesis models. The following main results are obtained: 1) The observations are well explained by extended bursts of star formation or a superposition of several bursts. Ages and burst durations are estimated. This naturally explains both the observed WR populations (including WN and WC stars) and the presence of red supergiants. 2) The fitted SEDs indicate that the stellar light suffers from a smaller extinction than that of the gas, confirming independent earlier findings. 3) All the considered observational constraints are compatible with a Salpeter IMF extending to masses >~ 40 Msun. Adopting a conservative approach we derive a LOWER LIMIT of Mup >~ 30 Msun for the Salpeter IMF. From more realistic assumptions on the metallicity and SF history we favour a lower limit Mup >~ 30-40 Msun, which is also in agreement with Hbeta equivalent width measurements of metal-rich HII regions in spiral galaxies indicating an upper mass cut-off of at least ~ 35 - 50 Msun. Steep IMF slopes (alpha >~ 3.3) are very unlikely. (abridged/modified abstract)
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