The discussion revolves around an inequality challenge involving two positive real numbers, \(a\) and \(b\), under the condition that \(a^3 + b^3 = a - b\). Participants are tasked with proving the inequality \(2(\sqrt{2}-1)a^2 - b^2 < 1\) or its variants.
Participants express disagreement regarding the form of the inequality, with some advocating for a strict inequality while others propose a non-strict version. There is no consensus on the correct formulation of the inequality.
The discussion highlights the potential for misunderstanding in the formulation of inequalities, particularly in the context of the given condition \(a^3 + b^3 = a - b\). The implications of the inequality's tightness remain unresolved.
it should be:If $a$ and $b$ are two positive real, and that $a^3+b^3=a-b$, prove that $2\left(\sqrt{2}-1\right)a^2-b^2<1$.anemone said:If $a$ and $b$ are two positive real, and that $a^3+b^3=a-b$, prove that $2\left(\sqrt{2}-1\right)a^2-b^2\le 1$.
anemone said:If $a$ and $b$ are two positive real, and that $a^3+b^3=a-b$, prove that $2\left(\sqrt{2}-1\right)a^2-b^2\le 1$.