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A distribution function is called strong unimodal if its composition with any unimodal distribution function is unimodal. The following theorem is proved: For a proper unimodal distribution F (x) to be strong unimodal, it is necessary and sufficient that the function F (x) be continuous, and the function log F' (x) be concave at a set of points where neither the right nor the left derivative of the function F (x) is equal to zero.
И. А. Ибрагимов (Sun,) studied this question.