My problem concerns the big-O error after transforming a constraint optimization problem. I'm not really familiar in this field so sorry for any incorrect notation/logic in my reasoning. I will first explain the steps I make and ask the question in the end.
Let $n$ be a positive integer. Consider the following optimization problem involving positive integers $a$ and $b$:
minimize $ab$ subject to the constraint $$\binom{a+b}{a}\geq n^{2\sqrt{b}}.$$
Since I do not know the value of $n$, I transform the problem by expressing the problem in terms of variables that depend on $n$. More precisely, I define $\alpha = a / (\lg n)^2$ and $\beta = b / (\lg n)^2$ ($\lg$ is base 2 logarithm). I use Stirling's approximation for the logarithm of a factorial:
$$\lg(k!) = k\lg k -k\lg e + O(\lg k).$$
Using this, I can rewrite (skipping a few of the steps in which certain parts cancel out) the logarithm of the binomial coefficient as follows:
\begin{align} \lg\binom{a+b}{a} &= \lg \binom{(\alpha + \beta)(\lg n)^2}{\alpha(\lg n)^2}\\ &=(\lg n)^2\left[(\alpha+\beta)\lg\left((\alpha + \beta)\right)-\beta\lg\beta -\alpha\lg\alpha \right]+ O\left(\lg((\alpha + \beta)(\lg n)^2)\right) -O\left(\lg(\beta(\lg n)^2)\right) -O\left(\lg(\alpha(\lg n)^2)\right) \end{align}
The bound on the RHS can be written as
$$\lg(n^{2\sqrt{b}}) = 2\sqrt{\beta (\lg n)^2}(\lg n) = 2\sqrt{\beta}(\lg n)^2.$$
Now, my idea was to 'ignore' the big-O terms and consider the constraint:
$$ (\lg n)^2\left[(\alpha+\beta)\lg\left((\alpha + \beta)\right)-\beta\lg\beta -\alpha\lg\alpha \right]\geq 2\sqrt{\beta}(\lg n)^2 $$
I can divide on both sides by $(\lg n)^2$ and obtain a constraint that does not explicitly depend on $n$. So I have the optimization problem
minimize $\alpha\beta$ subject to the constraint $$(\alpha+\beta)\lg\left((\alpha + \beta)\right)-\beta\lg\beta -\alpha\lg\alpha\geq 2\sqrt{\beta}.$$
This I can solve numerically for $\alpha$ and $\beta$ and obtain a solution to the original problem in terms of $n$.
My question now is, how to describe the error term of this solution? I used Stirling's approximation and ignored the 'big-O' terms. For example, say that I obtain $\alpha=\beta=0.5$ as a solution. This would mean $a=b=0.5(\lg n)^2 + O(\text{<function>})$, but which function, in which variable(s), would be inside the big-O term? I don't really have a clue. How do the errors involved in the approximation of the binomial coefficient propagate to the final result?
Perhaps I could group the three error terms in the approximation above and get
$$O\left(\lg\left(\frac{\alpha+\beta}{\beta\alpha}\right)-\lg((\lg n)^2)\right).$$
Then, writing it in terms of $a$ and $b$ would give
$$O\left(\lg\left((\lg n)^2\frac{a+b}{ba}\right)-\lg((\lg n)^2)\right)=O\left(\lg\left(\frac{a+b}{ba}\right)\right),$$
would this make sense?
Thanks in advance for any help!