Optimal Quadratic Refinements of Bernoulli’s Inequality and Some Related Results

Abstract

Bradley (2025) proved a quadratic strengthening of the classical Bernoulli inequality for integer exponents, and Andrusenko, Shevchuk and Wójcik (2025) established a convex-function equivalence theorem for three functions. This note collects a clean set of consequences and extensions. We prove that Bradley’s quadratic coefficient is optimal on its natural domain, extend the forward implication to k nonnegative functions in a pointwise form, establish the real-exponent quadratic inequality

                                          (1 + x)^a > 1 + ax + (a  – 1)x²

for every a > 2 and x > –1 with best possible coefficient a  – 1, derive quadratic strengthenings and numerical corollaries, show that no positive quadratic coefficient can hold globally for 1 < a < 2, and derive the corresponding operator inequalities for self-adjoint operators.