Bounding function of norms in constrained vector space

Question

$v$ is a vector of length $n$, where $v_1 = 1$ and every element $v_i \in [0,1]$

$w = \| v \|_1^1 = \sum_i |v_i| = \sum_i v_i$
$x = \| v \|_2^2 = \sum_i |v_i|^2 = \sum_i v_i^2$
$y = \| v \|_3^3 = \sum_i |v_i|^3 = \sum_i v_i^3$
$z = \| v \|_4^4 = \sum_i |v_i|^4 = \sum_i v_i^4$

Can you recommend a strategy for achieving a bound on

$$f = \frac{w z - x y + \sqrt{w^2 z^2 - 6 w x y z + 4 w y^3 + 4 x^3 z - 3 x^2 y ^2 }}{2 (w y - x^2)}$$

for all possible $v$ described above and where the denominator is nonzero: $(w y - x^2) \neq 0$ ? The nonzero denominator implies that the vector $v$ has an element $v_{i\neq 1} \in (0,1)$, which can be seen by assuming all elements are $\in \{0,1\}$ and yielding a contradiction. Thus it can be shown that when the denominator equals zero, the numerator also equals zero (because $w = x = y = z$). When I've simulated numerically (choosing values in $v_{2 \ldots n}$ uniformly in $[0,1]$ a million times), and have observed that $f$ is generally quite close to $1$.

Can you help me find a bound (even a loose bound) $f \in [lower(n), upper(n)]$? I've tried starting from the KKT criteria to define $v$ that maximizes (or, alternatively, minimizes) $f$, but the expression was so messy that Mathematica choked even when using a small vector length $n$.

Optimizing over $w, x, y, z$, which should each be $\in [0, n]$ and where $w \geq x \geq y \geq z$ (because, e.g. $v_i \geq v_i^2 \rightarrow w \geq x$); however, that is still not constrained enough, because it allows that the denominator can be close to zero when the numerator is nonzero (e.g., $w \approx x \approx y << z$. But there is no vector $v$ where $w \approx x \approx y << z$: $\| v \|_p^p$ is decreasing, concave up, meaning that $z - y < y - x$, and thus $z$ cannot jump quickly if $w \approx x \approx y$. But even with this additional constraint on $w,x,y,z$, I have not been able to find a bound on $f$.

I've also tried letting $\alpha$ denote an element of $v_{i\neq 1} \in (0,1)$ so that $w = 1 + \alpha + \| u \|_1^1$, $y=1 + \alpha^2 + \| u \|_2^2$, etc., where $u$ is a vector of length $n-2$ and all elements in $[0,1]$, and then maximizing (or minimizing) $f$ with respect to $w',x',y',z',\alpha$, where $w' = \| u \|_1^1, x' = \| u \|_2^2, \ldots$, but have also not yet found success.

Do you see appropriate additional constraint(s) that will adequately describe that $w,x,y,z$ behave as norms (without letting $v$ creep back into the problem)? Do you see any place where I can massage $f$ into some larger (or smaller for the lower bound) expression that permits a loose bound? Or, more generally, can you recommend a plan of attack?

In case you're interested, I've come to this problem through statistics; $f$ is an estimate that should ideally be 1, and I'm trying to bound the error of my estimate. Note that this problem is equivalent to

$$ max( roots( nullSpace \left[ \begin{array}{ccc} \|v\|_1^1 & \|v\|_2^2 & \|v\|_3^3 \\ \|v\|_2^2 & \|v\|_3^3 & \|v\|_4^4 \\ \end{array} \right] ) ) )$$

Thank you for any advice or ideas you have.

Answer 1

Experimentally, $$0.7 <\ f \ \le\ 1.$$

The upper bound is rigorous. First we show $wy-x^2=\sum_{i<j}v_i v_j (v_i-v_j)^2 \ge 0$. Then we can use this to simplify the upper bound to $(wy-x^2)+(xz-y^2)-(wz-xy) = \sum_{i<j} v_i v_j(1-v_i)(1-v_j)(v_i-v_j)^2 \ge 0$. The extreme case is $v$ of any length of the form $$v=(1,1/2,\ldots,1/2),\ f=1.$$

The lower bound is more conjectural. Experimentation suggests looking at $v$'s of the form $(1,a,b,\ldots,b)$. For these $v$, we can show that the value of $f$ approaches $b$ as $v$ grows longer, so long as $(1-b)^3 \le a(b-a)^3$. We minimize $b$ subject to that constraint with $a=0.176$, $b=0.704$. Those values give the apparently extreme case where $v$ is long and of the form $$v=(1,\,0.176,\,0.704,\,\ldots,\,0.704),\ f=0.704.$$