Bounding derivatives of the sinc function

The sinc function is defined either as sin(x)/x or as sin(πx)/πx. We’ll use the former definition here because we’ll cite a paper that uses that definition.

Here’s a plot of the sinc function and its first two derivatives.

Thomas Grönwall proposed a problem to the American Mathematical Monthly in 1913 [1] bounding the derivatives of the sinc function:

\left| \frac{d^n}{dx^n} \frac{\sin x}{x}\right| \leq \frac{1}{n+1}

Seven years later, Dunkel gave an elegant proof. Perhaps Grönwall had a proof and was proposing his inequality as a challenge, or maybe it was a conjecture at the time he published it. In any case, the proof by Dunkel is very nice. He sets

y = sin(x)/x

and repeatedly differentiates both sides of the equation

xy = sin(x).

See the details in Dunkel’s solution.

I don’t know of an application of Grönwall’s inequality offhand. But the sinc function is common in signal processing, and so maybe his inequality has applications there.

(By “Grönwall’s inequality” I mean the inequality above. There is another theorem also known as Grönwall’s inequality that is commonly applied in differential equations.)

[1] Gronwall, T. H. (1913). Problem 339. Amer. Math. Monthly. 20: 196.

[2] Dunkel, O., (1920). Solution to Problem 339. Amer. Math. Monthly. 27: 81–85.