I am referring to this paper, p. 22.
On that page, from the inequality
$$ \begin{align*} &\mathbb{H}(f^{n+1})-\mathbb{H}(f^n)+\Delta t\left(\frac{1}{\varepsilon^2}-\eta\bar{m}_2\right)\Vert f^{n+1}-\rho^{n+1}\mathcal{M}\Vert_{2,\gamma}^2+\Delta t\eta\underline{m}_2\Vert\rho^{n+1}\Vert_2^2\\ &\leq\Delta t\eta\left(\sqrt{\overline{m}_4-\underline{m}_2^2}+\frac{C_P\sqrt{\overline{m}_2}}{\varepsilon}\right)\Vert f^{n+1} - \rho^{n+1}\mathcal{M}\Vert_{2,\gamma}\Vert\rho^{n+1}\Vert_2\tag{1} \end{align*} $$ the authors conclude that for any $\eta\in (0,\eta_2)$ with $$ \eta_2:=\frac{\underline{m}_2}{(\sqrt{\overline{m}_4-\underline{m}_2^2}+C_P\sqrt{\overline{m}_2})^2+\underline{m}_2\overline{m}_2}\tag{2} $$ and $\varepsilon\in (0,1)$ one has $$ \mathbb{H}(f^{n+1})-\mathbb{H}(f^n)+\Delta t K(\eta)(\Vert f^{n+1}-\rho^{n+1}\mathcal{M}\Vert_{2,\gamma}^2+\Vert\rho^{n+1}\Vert_2^2)\leq 0\tag{3} $$ with $$ K(\eta)=\frac{1}{2}\min(1-\eta\overline{m}_2,\eta\underline{m}_2).\tag{4} $$
Could somebody please explain to me how to deduce $(3)$?
I am trying for hours, but I simply do not get it. Is there some trick behind it?
I guess its a pure algebraic deduction and one does not need to know how all the notations appearing here are actually defined (this is why I omit the various definitions here...).
Actually, I have no concrete idea how to start.
To shorten the notation a little bit, I set $$ x:=\Vert f^{n+1}-\rho^{n+1}\mathcal{M}\Vert_{2,\gamma},\qquad y:=\Vert\rho^{n+1}\Vert_2. $$
Since $(1)$ is equivalent to $$\small{ \begin{align*} &\mathbb{H}(f^{n+1})-\mathbb{H}(f^n)+\Delta t\left(\frac{1}{\varepsilon^2}-\eta\bar{m}_2\right)x^2+\Delta t\eta\underline{m}_2y^2-\Delta t\eta\left(\sqrt{\overline{m}_4-\underline{m}_2^2}+\frac{C_P\sqrt{\overline{m}_2}}{\varepsilon}\right)xy\leq 0 \end{align*}} $$
I guess that, in order to show $(3)$, one has to prove that, for $\eta\in (0,\eta_2)$ and $\varepsilon\in (0,1)$, $$ \begin{align*} K(\eta)(x^2+y^2)\leq\left(\frac{1}{\varepsilon^2}-\eta\bar{m}_2\right)x^2+\eta\underline{m}_2y^2-\eta\left(\sqrt{\overline{m}_4-\underline{m}_2^2}+\frac{C_P\sqrt{\overline{m}_2}}{\varepsilon}\right)xy \end{align*} $$
(I did not manage to continue from here, however. It nearly drives me crazy.)