A Gronwall-type Trigonometric Inequality

TL;DR

This paper establishes a new inequality bounding the derivatives of the cosine of the square root function, extending it through analytic continuation, and provides a generalization of this inequality.

Contribution

It introduces a novel Gronwall-type inequality for derivatives of a special function and generalizes it via analytic continuation.

Findings

Proved the derivative bound for cos(√x) for all x>0 and n≥0.

Derived a natural generalization involving analytic continuation.

Enhanced understanding of inequalities related to special functions.

Abstract

We prove that the absolute value of the $n$th derivative of $\cos(\sqrt{x})$ does not exceed $n!/(2n)!$ for all $x>0$ and $n = 0,1,\ldots$ and obtain a natural generalization of this inequality involving the analytic continuation of $\cos(\sqrt{x})$.