Question: How to determine convexity of this function?

This question is related to the Post convexity

Let us consider the expression

f := log[x](1+(x^a-1)*(x^b-1)/(x-1));
Does it define a convex function on the interval 0..1 and on the interval 1..infinity if the parameters a>0, a< 1, b >0, b <1?

My try is



At the same time I have got problems in the general case. For example,

Error, (in @) too many levels of recursion


That's all right if



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