Moroccan Journal of Algebra and Geometry with Applications
A new look at an old inequality
David E. Dobbs
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320
Pages 196-285 | Received 04 April 2022, Accepted 18 July 2022, Published 06 December 2022
Abstract
For motivation and to enhance accessibility, we begin by reviewing some well known proofs that |x + 1/x| \geq 2 for all nonzero real numbers x and also note that there exist nonzero complex numbers z such that |z + 1/z| < 2. Then, for each B > 0, we determine the nonzero complex numbers z such that |z + 1/z| = B (resp., |z + 1/z| < B). Identifying the complex plane with \mathbb{R}^2 via the usual Argand diagram, we show that this set has Jordan content 0 (resp., give a formula for the Jordan content of this set). This study leads to a sense in which, for a random nonzero complex number z such that |z| \leq 1, the probability that |z + 1/z| < 2 is 2/\pi (and hence is greater than 0.63). Various portions of this paper could be used as enrichment material in courses ranging from precalculus to abstract algebra, advanced calculus, real analysis and general topology.
Keywords: Inequality, complex number, absolute value, unit disk, Fundamental Theorem of Calculus, Mean Value Theorem, L’Hôpital’s Rule, accumulation point, C^1 function, geometric probability, Jacobian, polynomial ring.
MSC numbers: Primary 00-01; Secondary 33B10, 26A06, 26A15, 54A99.
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