I'm looking specifically at the optimization problem
$$
\begin{align\*}
\text{maximize: }& n - \frac{\|\lambda\|\_1^2}{\|\lambda\|\_2^2}\\
\text{subj. to: }& \lambda \succeq \epsilon\mathbf{1}
\end{align\*}
$$
over $\mathbb{R}^n$ where $\epsilon\geq 0$ is some constant. This is equivalent to the problem
$$
\begin{align\*}
\text{minimize: }& \frac{\|\lambda\|\_1}{\|\lambda\|\_2}\\
\text{subj. to: }& \lambda \succeq \epsilon\mathbf{1}
\end{align\*}
$$
which at least looks cleaner, and is in fact a [concave fractional program](https://en.wikipedia.org/wiki/Fractional_programming#Concave_fractional_programs) (at least if $\epsilon$ is positive). 


**Question:** Is there a way to bound the optimal value of the original objective function? Preferably can we show that for some small (in $n$) $\epsilon$ that the optimal $n - \frac{\|\lambda\|\_1^2}{\|\lambda\|\_2^2} < 1$?


**When $\epsilon=0$:** In this case we can rely on the tight bound $\|x\|\_2 \leq \|x\|\_1$ to have an optimal value of $n - \frac{\|\lambda^\star\|\_1^2}{\|\lambda^\star\|\_2^2} = n-1$.



Tighest Gap $\|x\|_1/\|x\|_2$ between $\ell^1$ and $\ell^2$ norms
(EDIT: changing this to at least solve the stated problem...)


We first look for the optimal vector for fixed $\|\lambda\|\_1=M$. Hence the problem is: maximize $\|\lambda\|\_2^2$ s.t. $\|\lambda\|\_1=M$, $\lambda\geq \varepsilon\mathbf{1}$.


The standard trick to solve this kind of inequalities is called *unsmoothing*, and goes as follows. Start from any feasible point, and assume without loss of generality that $\lambda\_1\geq \lambda\_j$ for each $j$.


Now, a simple computation shows that replacing $\lambda\_1,\lambda\_j$ with $\lambda\_1+\delta,\lambda\_j-\delta$ increases the objective function. (Remark: this is essentially equivalent to noticing that $t^2 + (\lambda\_1+\lambda\_j-t)^2$ is convex in $t$, so its maximum must be at one end of the domain).


Hence, with $n-1$ of these replacement steps, you arrive to $(M-(n-1)\varepsilon,\varepsilon,\varepsilon,\dots,\varepsilon)$, and your objective function has increased at each step. This proves that the maxima are $(M-(n-1)\varepsilon,\varepsilon,\varepsilon,\dots,\varepsilon)$ and its cyclic permutations.


Now we get 
$$
\frac{\|\lambda\|\_1}{\|\lambda\|\_2} \leq \frac{M}{\sqrt{(M-(n-1)\varepsilon)^2+(n-1)\varepsilon^2}}
$$
which is now a univariate problem.


