For every \(q\ge 2\) and \(x\in \mathbb R\), \(x^2\le |1+x|^q-1-qx\).

For every \(m\in \mathbb N\) and \(x\in \mathbb R\) with \(|x|\le 1\), \((1+x)^m-1-mx\le (2^m-1-m)x^2\).

For every \(q\ge 2\), an integer \(m\ge q\), and \(|x|\le 1\), \(|1+x|^q-1-qx\le 2^m x^2\).

For every real \(q\ge 2\), there is \(M\gt 0\) such that \(|1+x|^q-1-qx\le M(x^2+|x|^q)\) for every real \(x\).

For every \(q\ge 0\) and \(x\in \mathbb R\), \(|x|^q\le 2^q(|1+x|^q+1)\).

For every \(q\ge 2\) and \(x\in \mathbb R\), \(|x|^q\le 2^q(q+3)(|1+x|^q-1-qx)\).

For every \(q\ge 2\) there is \(0\lt c\le 1\) such that \(|1-cz|^q\le |1+z|^q-q(1+c)z\) for every real \(z\).

For every \(q\ge 2\) there is \(0\lt c\le 1\), with \(c=2/5\) when \(q=4\), such that \(|1-cz|^q\le |1+z|^q-q(1+c)z\) for every real \(z\).

Let \(q\gt 0\) and \(0\le c\le 1\) satisfy \(|1-cz|^q\le |1+z|^q-q(1+c)z\) for every real \(z\). Then for all real \(a,x\), \(|a-cx|^q\le |a+x|^q-[q(1+c)|a|^q/a]x\), with the quotient interpreted as zero when \(a=0\).

Let \((\Omega ,\mu )\) be a probability space, \(q\ge 2\), and \(0\le c\le 1\) satisfy \(|1-cz|^q\le |1+z|^q-q(1+c)z\) for every real \(z\). If \(X\in L^q(\mu )\) is real with \(\int X\, d\mu =0\), then for every \(a\in \mathbb R\), \(\| a-cX\| _{L^q(\mu )}\le \| a+X\| _{L^q(\mu )}\).