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As whuber notes, this is the class of distributions with log-convex probability density functions, which are a bit less studied than distributions with log-concave densities.

A good free resource on some of the "named" distributions that have log-convex densities is

[Mark Bagnoli and Ted Bergstrom. "Log-concave Probability and its Applications" 2004][1]


It focuses on log-cocavity but it also contains results on log-convexity. We see that the OP's assertion that "Weibull, normal, log-normal, exponential, gamma" have log-convex densities is incorrect: for example, the normal and the exponential distribution have _log-concave_ densities, while Weibull and gamma have also log-concave densities for some of their incarnations, while for others they have indeed log-convex densities. The log-normal has a density that is neither log-concave, nor log-convex.

Another resource that examines log-convexity more abstractly and also more thoroughly is 

[An, M. Y. (1998). Logconcavity versus logconvexity: a complete characterization. Journal of Economic theory, 80(2), 350-369.][2]


  [1]: http://works.bepress.com/ted_bergstrom/98/
  [2]: http://www.researchgate.net/profile/Mark_An/publication/4977452_Logconcavity_versus_Logconvexity_A_Complete_Characterization/links/540a343c0cf2d8daaabfa20c.pdf