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I think the answer to both is negative. By a result by Carleson, entire real functions are dense in $C(\mathbb{R},\mathbb{R})$ with the Withney topology, so there is an entire $f$ such that $$1/x + \sin(e^x) /x^2 < f(x) < 1/x + \sin(e^x)/x^2 +1/x^3$$ for all $x>0$; this clearly forces $f'(x)$ to be unbounded for $x\to+\infty$. Analogously for the other question.

I think the answer to both is negative. By a result by Carleman, entire real functions are dense in $C(\mathbb{R},\mathbb{R})$ with the Withney topology, so there is an entire $f$ such that $$1/x + \sin(e^x) /x^2 < f(x) < 1/x + \sin(e^x)/x^2 +1/x^3$$ for all $x>1$; this clearly forces $f'(x)$ to be unbounded for $x\to+\infty$. Analogously for the other question.

I think the answer to both is negative. By a result by Carleson, entire real functions are dense in $C(\mathbb{R},\mathbb{R})$ with the Withney topology, so there is an entire $f$ such that $$1/x - \sin(e^x)/x^2 < f(x) < 1/x + \sin(e^x)/x^2$$ for all $x>0$; this clearly forces $f'(x)$ to be unbounded for $x\to+\infty$. Analogously for the other question.

I think the answer to both is negative. By a result by Carleson, entire real functions are dense in $C(\mathbb{R},\mathbb{R})$ with the Withney topology, so there is an entire $f$ such that $$1/x + \sin(e^x) /x^2 < f(x) < 1/x + \sin(e^x)/x^2 +1/x^3$$ for all $x>0$; this clearly forces $f'(x)$ to be unbounded for $x\to+\infty$. Analogously for the other question.

I think the answer to both is negative. By a result by Carleson, entire real functions are dense in the Withney topology, so there is an entire $f$ such that $$1/x - \sin(e^x)/x^2 < f(x) < 1/x + \sin(e^x)/x^2$$ for all $x>0$; this clearly forces $f'(x)$ to be unbounded for $x\to+\infty$. Analogously for the other question.

I think the answer to both is negative. By a result by Carleson, entire real functions are dense in $C(\mathbb{R},\mathbb{R})$ with the Withney topology, so there is an entire $f$ such that $$1/x - \sin(e^x)/x^2 < f(x) < 1/x + \sin(e^x)/x^2$$ for all $x>0$; this clearly forces $f'(x)$ to be unbounded for $x\to+\infty$. Analogously for the other question.