A Simple Proof of an Inequality for the Complete Elliptic Integral of the First Kind Harun Abdul Rohman SMPN 2 Cilengkrang, Bandung Regency, West Java, Indonesia harun88pltg@gmail.com April 24, 2025 Abstract This article presents a simple proof of inequalities for the complete elliptic integral of the first kind through symmetry arguments and interval reduction techniques. Fur- thermore, it provides a geometric interpretation of the bounds for this integral, offering intuitive insights into its analytic behavior. Keywords: Complete elliptic integral, First kind elliptic integral, Symmetry argu- ments, Inequalities. 1 Introduction In his article, Jameson [1] introduces the integral I (a, b)=4 Z π 2 0 1 p a 2 cos 2 (t)+ b 2 sin 2 (t) dt (1) where a ≥ b> 0, which frequently appears alongside elliptic integrals in the context of ellipses. He establishes the inequality π a + b ≤ I (a, b) ≤ π 2 √ ab (2) using Gauss’s identity I (a, b)= 2π AGM(a,b) , where AGM (a, b) is the arithmetic-geometric mean of a and b, and satisfies √ ab ≤ AGM (a, b) ≤ a+b 2 [2–4]. The integral I (a, b) can also be written in terms of the complete elliptic integral of the first kind K (k), as I (a, b)= 4 a Z π 2 0 1 p 1 − k 2 sin 2 (t) dt = 1 a K (k), where k 2 =1 − b 2 a 2 , so that aI (a, b)= K (k) ⇒ πa a + b ≤ K (k) ≤ πa 2 √ ab . (3) 1 https://doi.org/10.33774/coe-2025-dq13h ORCID: https://orcid.org/0000-0002-3639-2910 Content not peer-reviewed by Cambridge University Press. License: CC BY-NC-ND 4.0