Inequalities for Integer and Fractional Parts

In this paper we present 43 new inequalities related to integer part and fractional part.

💡 Research Summary

The paper “Inequalities for Integer and Fractional Parts” introduces a comprehensive collection of forty‑three novel inequalities that involve the integer part (floor function) ⌊x⌋ and the fractional part {x}=x−⌊x⌋ of real numbers. The authors begin by reviewing the basic algebraic and analytic properties of these two functions, emphasizing their monotonicity, periodicity, and sub‑additivity, and they point out that while many elementary relations such as ⌊x⌋≤x≤⌊x⌋+1 are well known, far fewer results address non‑linear combinations of integer and fractional components.

The first set of results concerns single‑variable inequalities that go beyond the linear bounds. For any real x with fractional part t∈