How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?

Solution 1:

By the result of [1], we have $$\left|\mathrm{Si}(y) - \frac{\pi}{2}\right| \le \frac{1}{y}, \ y > 0.\tag{1}$$ Thus, we have, for $z > 0$, $$\frac{\pi}{2} - \frac{1}{z} \le \mathrm{Si}(z) \le \frac{\pi}{2} + \frac{1}{z}$$ and $$\frac{\pi}{2} - \frac{1}{\mathrm{e}z} \le \mathrm{Si}(\mathrm{e}z) \le \frac{\pi}{2} + \frac{1}{\mathrm{e}z}.$$ Thus, we have, for $z > 0$, $$-1 - \frac{1}{\mathrm{e}} \le z(\mathrm{Si}(\mathrm{e}z) - \mathrm{Si}(z)) \le 1 + \frac{1}{\mathrm{e}}.$$ Also, $1 + \frac{1}{\mathrm{e}} \approx 1.367879441$. We are done.

Reference

[1] Upper bound for the sine integral

Remark: The following stronger inequality holds, which implies (1): $$\left|\mathrm{Si}(y) - \frac{\pi}{2}\right| \le \arctan \frac{1}{y}, \ y > 0.$$ See: Graham Jameson, Nick Lord and James McKee, An inequality for Si(x), Math. Gazette 99 (2015). https://www.maths.lancs.ac.uk/jameson/siineqnotes.pdf

Solution 2:

This question has received some answers on MathOverflow.

• Fyodor Petrov solves with integration by parts
• Iosif Pinelis suggests a method that allows arbitrarily accurate bounds with interval arithmetic