I am trying to define a function $\beta(\delta)$ which will be a polynomial of $k,H,d$ and $1/\delta$ satisfying $\beta(\delta) = \tilde{O}(H\sqrt{d})$ where $\tilde{O}$ is similar to Big-O notation that ignores logarithmic terms. The only other requirement is that $\beta(\delta)$ needs to dominate the following term.
$\text{RHS} = 2H\sqrt{d}\Bigl[\frac{1}{2}\log(k+1) + \log3\big\{2\sqrt{k^3} + (\frac{k}{H}\sqrt{\beta(\delta)} + \frac{k}{H}\sqrt{d} + k\sqrt{d})c\sqrt{\log(d/\delta)}\big\} + \log(1/\delta)\Bigr]^{1/2}$
that is we want $\beta(\delta) \geq \text{RHS}$.
Note that $\text{RHS}$ has $\beta(\delta)$ within it. I am not sure how to approach this and would really appreciate any help.
I am essentially trying to do something like the following from this paper https://arxiv.org/pdf/1911.00567.pdf
