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So I need to prove something or see if its true and I don't know how to write in nice math text because this is my first question, so please bear with me

We have a CDF $F_h(s)$ for $0\le s\le 1$. The corresponding density $f_h$ is increasing, convex, and Log-concave. Prove that if there exists $t\in(0,\frac12)$ s.t. $$ \frac{f_h'(t)}{f_h(t)+1} > \frac{f_h'(1-t)}{f_h(1-t)+1} $$ then for every $t'\in(t,\frac12)$ we also have $$ \frac{f_h'(t')}{f_h(t')+1} > \frac{f_h'(1-t')}{f_h(1-t')+1} $$

Any help would be greatly appreciated. Thanks

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  • $\begingroup$ Do you have a source for the question? $\endgroup$ – wolfies Oct 26 '14 at 12:25
  • $\begingroup$ no its my question...from research... $\endgroup$ – Daniel Oct 26 '14 at 18:55

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