On general families of cyclic quadratic inequalities

Abstract

This paper introduces general families of parameterized cyclic quadratic inequalities in three positive real variables. We first establish eight- and nine-parameter algebraic inequalities, both of which are subsequently generalized to functional forms using convex and non-decreasing functions. We then formulate a broader ten-parameter weighted cyclic inequality that relaxes the monotonicity requirement, holding for any arbitrary convex function. By specializing these parameters and selecting standard convex functions, we derive numerous explicit inequalities, demonstrating the capacity to unify classical results and generate new bounds.

1. Introduction

While many specific cyclic quadratic inequalities exist, comprehensive theoretical frameworks remain rare. Typically, these inequalities are approached on a case-by-case basis using ad hoc algebraic methods. Developing parameterized theorems that unify broad families of such inequalities is therefore an important objective. To address this gap, we establish a unified framework of parameterized cyclic inequalities in three positive real variables. Specifically, we begin by introducing eight- and nine-parameter algebraic inequalities, which we subsequently generalize to functional forms via convex and non-decreasing functions. Next, we formulate a broader ten-parameter weighted inequality that relaxes the monotonicity requirement, holding for any arbitrary convex function. Finally, we demonstrate the utility of these theorems by recovering classical inequalities and generating new bounds as special cases.

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Cite this article

Chesneau, C. (2026). On general families of cyclic quadratic inequalities. Asia Mathematika, 10(2), 136–154. https://doi.org/10.5281/zenodo.23123890

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