If $2(1-Cs)\,I(s)\le s\,I'(s)$ on $(0,S)$, then $e^{2Cs}I(s)/s^2$ is nondecreasing there

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If $2(1-Cs)\,I(s)\le s\,I'(s)$ on $(0,S)$, then $e^{2Cs}I(s)/s^2$ is nondecreasing there · Prove2Me