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I want to prove this for any $x > 0$ and $b > a > 0$:

$$ \frac{2}{\pi} (1-\frac{a}{b})<\sup|\frac{\sin(ax)}{ax} - \frac{\sin(bx)}{bx}|<4(1-\frac{a}{b}) $$

I tried the derivation to find the maximum value, then I tried to insulate the $\frac{\sin(ax)}{ax}$ and I got this:

$$\sup\left|\frac{\sin(ax)}{ax} \right|\cdot \left|1 - \frac{a}{b}\cdot \frac{\sin(bx)}{\sin(ax)}\right| $$

I think this may be the key to prove that Notice that the relation is true, I used Geogebra to plot the function with many configuration of a and b, and I get this relation true any help, please.

Gary
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Mohamed Farouk HASNAOUI
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1 Answers1

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We can prove that $$ 1 - \frac ab \le \sup_{x > 0}\left|\frac{\sin(ax)}{ax} - \frac{\sin(bx)}{bx}\right| \le 2\left(1 - \frac ab \right) $$ for $0 < a < b$, which is a better estimate.

Proof of the lower bound: For $0 < a < b$ and $x = \pi/b$ is $$ \left|\frac{\sin(ax)}{ax} - \frac{\sin(bx)}{bx}\right| = \frac{\sin(\pi a/b)}{\pi a/b} \ge 1 - \frac ab \, . $$

For the last inequality, see for example sinus estimation by quadratic polynomial or How prove this inequality $\pi<\frac{\sin{(\pi x)}}{x(1-x)}\le 4$ .

Proof of the upper bound: Let $0 < u < v$. We have $$ \left|\frac{\sin(u)}{u} - \frac{\sin(v)}{v}\right| \le \left|\frac{\sin(u)}{u} - \frac{\sin(u)}{v} \right|+\left|\frac{\sin(u)}{v}- \frac{\sin(v)}{v}\right| \\ = |\sin(u)| \frac{v-u}{uv} + \frac{|\sin(u)-\sin(v)|}{v} \, . $$ Using $|\sin(u)| \le |u|$ and $|\sin(u)-\sin(v) |\le |u-v|$ it follows that $$ \left|\frac{\sin(u)}{u} - \frac{\sin(v)}{v}\right| \le 2\frac{v-u}{v} = 2 \left(1 - \frac uv \right) \, . $$ For $0< a < b$ and $x > 0$ we can now set $u=ax$ and $v=bx$ and conclude that $$ \left|\frac{\sin(ax)}{ax} - \frac{\sin(bx)}{bx}\right| \le 2 \left(1 - \frac ab \right) \, . $$

Martin R
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