Show that $\frac{2}{\pi}\left(1-\frac{a}{b}\right)\leq\sup_{x>0}|\frac{\sin (ax)}{ax}-\frac{\sin (bx)}{bx}|\leq 4\left(1-\frac{a}{b}\right)$

Asked Apr 06 '22 at 17:33

Active Apr 07 '22 at 02:25

Viewed 49 times

Exercise:

Let $a~,~b$ be a real number such that : $b>a>0$ then prove that
$$\frac{2}{\pi}\left(1-\frac{a}{b}\right)\leq\sup_{x>0}|\frac{\sin (ax)}{ax}-\frac{\sin (bx)}{bx}|\leq 4\left(1-\frac{a}{b}\right).$$

I was trying to study the function

$$f(x)=\frac{\sin (ax)}{ax}-\frac{\sin (bx)}{bx}.$$ But no results! I need just an idea or ideas to start solving this impressive problem.

edited Apr 07 '22 at 02:25

Gary

31,845

asked Apr 06 '22 at 17:33

Ellen Ellen

2,319

1

A sharper bound was recently proven here: https://math.stackexchange.com/q/4416911/42969 – Martin R Apr 06 '22 at 17:49
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Where does the problem come from? It was also asked yesterday: https://math.stackexchange.com/q/4420626/42969, so this is the third time in one week that this comes up. – Martin R Apr 06 '22 at 17:50

Show that $\frac{2}{\pi}\left(1-\frac{a}{b}\right)\leq\sup_{x>0}|\frac{\sin (ax)}{ax}-\frac{\sin (bx)}{bx}|\leq 4\left(1-\frac{a}{b}\right)$

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