Let \(q\gt 2\) and \(0\lt \lambda \lt 1/2\). Then \(0\lt \rho _*(q,\lambda )\lt 1\). Here \(\rho _*(q,\lambda )=\sqrt{\sinh (\ell /q)/\sinh (\ell /(q/(q-1)))}\) with \(\ell =\log ((1-\lambda )/\lambda )\).

Let \(p\in \mathbb {R}\) and \(0\lt \alpha \lt \beta \). Then \((\alpha ^p\beta ^{2-p}-\alpha ^{2-p}\beta ^p)/(\alpha ^2-\beta ^2)=\sinh ((p-1)\log (\beta /\alpha ))/\sinh (\log (\beta /\alpha ))\).

Let \(1\lt p\lt 2\) and \(0\lt \alpha \lt \beta \). Then \(0\lt (\alpha ^p\beta ^{2-p}-\alpha ^{2-p}\beta ^p)/(\alpha ^2-\beta ^2)\lt 1\).

Let \(1\lt p\lt 2\), \(0\lt \alpha \lt \beta \), and \(\alpha ^p+\beta ^p=1\). Then \((\alpha ^p\beta ^{2-p}-\alpha ^{2-p}\beta ^p)/(\alpha ^2-\beta ^2)=\rho _*(p/(p-1),\alpha ^p)^2\). Here \(\rho _*(q,\lambda )=\sqrt{\sinh (\ell /q)/\sinh (\ell /(q/(q-1)))}\) with \(\ell =\log ((1-\lambda )/\lambda )\).

The hyperbolic tangent \(x\mapsto \tanh x\) is concave on \([0,\infty )\).

For \(0\le a\le 1\), the function \(u\mapsto \sinh (au)/\sinh u\) is nonincreasing on \((0,\infty )\).

Let \(q\gt 2\) and \(0\lt \lambda \le \gamma \lt 1/2\). Then \(\rho _*(q,\lambda )\le \rho _*(q,\gamma )\). Here \(\rho _*(q,\lambda )=\sqrt{\sinh (\ell /q)/\sinh (\ell /(q/(q-1)))}\) with \(\ell =\log ((1-\lambda )/\lambda )\).

Let \(0\le a\le 1\) and \(u\ge 0\). Then \(\sinh (au)\le a\sinh u\).

Let \(q\gt 2\) and \(0\lt \lambda \lt 1/2\). Then \(\rho _*(q,\lambda )^2\le 1/(q-1)\). Here \(\rho _*(q,\lambda )=\sqrt{\sinh (\ell /q)/\sinh (\ell /(q/(q-1)))}\) with \(\ell =\log ((1-\lambda )/\lambda )\).