@fedja: Thank you, that's quite enlightening.

Altogether, very nice.

Fedja, thank you. Because $d e^x / dx = e^x$, you can show that $e^x$ is positive by asking what would happen at a zero crossing. I've tried (unsuccessfully) to analyze $f'(x)$ with the hope of doing something similar. Your proof strikes me as having the same flavor as that sort of manipulation. Could you offer any motivation for how you came up with your integral "operator"?

GH, thanks to you and Noam (and to de Bruijn) for this nice piece of analysis.