Deriving inequalities from other inequalities

Question

My questions come from the proof of Theorem 5.14 in section 5.7 of Boucheron, Lugosi, and Massart - Concentration inequalities. My first question can be stated as follows:

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Suppose for positive numbers $V,y,\Delta,x,\delta >0$ we have $V \leq y^{-1}\left[2\Delta y^{-1} + \sqrt{2x} + \sqrt{2\delta}\right].$ Then if we define $$y^2 = 2K\epsilon^2\left[\left(\sqrt{x}+\sqrt{\delta}\right)^2 + \epsilon^{-1}K^{-\frac 12}\Delta + \sqrt{\delta\epsilon^{-1}K^{-\frac 12}\Delta} \right] \tag{1}\label{1}$$ for $K >1$ and $\epsilon >0$, then the authors claimed that the inequality for $V$ implies that $\epsilon V \leq K^{-\frac 12}$. To be honest, I have no clue as to the motivation of defining $y$ according to \eqref{1}. If I want to enforce the relation $V \leq K^{-\frac 12}/\epsilon$, I will just set up the relation $$y^{-1}\left[2\Delta y^{-1} + \sqrt{2x} + \sqrt{2\delta}\right] = K^{-\frac 12}/\epsilon,$$ which defines a quadratic equation in the variable $y^{-1}$ and whose solution can be explicitly found out. But this idea does not lead to the suggested expression for $y$….

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My second question is again regarding details. Setting $\delta := \left((4\pi)^{-\frac 12} + \sqrt{z}\right)^2$ for fixed $z > 0$. The authors mentioned that by using repeatedly the elementary inequality $2ab \leq \theta a^2 + \theta^{-1}b^2$, we can derive the following upper bound from the identity/definition \eqref{1} (recall that $K >1$ is fixed): $$K^{-\frac 12}y^2 \leq 2K\epsilon^2\left[\epsilon^{-1}\Delta + x + \sqrt{\epsilon^{-1}\Delta x} + \frac{2}{\sqrt{K} - 1}\left(\frac{1}{2\pi} + 2z\right)\right]. \tag{2}\label{2}$$ I tried a lot but still can not figure out the right way to arrive at \eqref{2} from \eqref{1}, so any help is appreciated!

Answer 1

If $y^2$ is defined by (1) and $$\epsilon = \frac{168}{97},x= \frac{1168561}{2916},z= \frac{121}{8281},\Delta = \frac{1}{1000},K= \frac{10201}{9604},$$ then the difference between the right-hand side of inequality (2) and its left-hand side is $-2.5248\ldots<0$, so that (2) fails to hold.

(What might be interesting, though, is that, for $y^2,\epsilon,x,z,\Delta,K$ as above, the values of the right-hand side and left-hand side are relatively close to each other: $2635.1\ldots$ and $2637.6\ldots$.)

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Also, if $y^2$ is defined by (1), $y>0$, $V=y^{-1}\left(2\Delta y^{-1} + \sqrt{2x} + \sqrt{2\delta}\right)$ and $$K= \frac{602}{61},x= \frac{7936}{57},\delta = \frac{34618}{41},\epsilon = \frac{6}{73},\Delta = \frac{37534}{29},$$ then the ratio of the left-hand side of inequality $$\epsilon V \le K^{-\frac 12} \tag{*}$$ to its right-hand side is $1.2496\ldots>1$, so that (*) fails to hold, too.