Answer by Christian Remling for A strange condition of convexity?
There is no such function. (So, technically, the answer to the question is yes.)In terms of $g=f'/f$, the inequality becomes $g\ge |1+g^2+g'|$ or $|g'/g+g+1/g|\le 1$, at least when $g>0$. This shows...
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During my research, I come across this question.Let $f \in C^2(\mathbb R, \mathbb R_+^*)$ with $\forall x \in\mathbb R, f'(x) \geq |f''(x)+f(x)|$.Is it true that $\forall x \in \mathbb R, f''(x) \geq 0$ ?
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