Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that $$ f’’(x) \leq 0\quad \text{and} \quad (af)’’(x) \leq 0$$ for all $x \in I$?
On existence of a convex function
Ali
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