What K value minimizes this function I got a function $$ nK^{\frac{-1}{\beta}}\log{K} + ZK\log{n} $$ where $K$ and $n$ are integers and $Z &gt; 0$.
I followed the typical way of differentiating the function and equating to zero.  But I couldn't find a closed form.  I got the following result
$$
\frac{Z\log{n}}{n} = \frac {\frac{\log{K}}{\beta}-1}{K^{\frac{1+\beta}{\beta}}}
$$
I am not sure how can I go from here.
The last equation you wrote gave me an idea for simplifications.
For simplifications, let $$K=x^{\frac{\beta }{\beta +1}}$$ which makes the function to be
$$f(x)=x^{-\frac{1}{\beta +1}} \left(\frac{\beta  n}{\beta +1} \log (x)+x Z \log(n)\right)$$
$$f'(x)=\frac{\beta  x^{-\frac{\beta +2}{\beta +1}} ((\beta +1) (x Z \log (n)+n)-n \log(x))}{(\beta +1)^2}$$
So, after simplifications, we look for the zero of
$$g(x)=(\beta +1) (x Z \log (n)+n)-n \log (x)$$ The only analytical solution of this equation express in terms of Lambert function
$$x=-\frac{n\,\,W(t)}{(\beta +1) Z \log (n)} \quad \text{where} \qquad t=-\frac{e^{\beta +1} (\beta +1) Z \log (n)}{n}$$ Then $K$ from $x$.
It remains to prove that this solution is a minimum. So,
$$f''(x)=\frac{x^{-\frac{1}{\beta +1}-2} (\beta  (\beta +2) n \log (x)-\beta  (\beta +1)
   ((\beta +3) n+x Z \log (n)))}{(\beta +1)^3}$$ I have not been able to prove it.
Let us try using $n=56789$, $Z=\pi$  and $\beta=\log(10)$. It would give $K=10.4097$. But, this is a maximum.
So, if you look for a minimum value of the function other than $K=0$, we need to use the second branch of Lambert function, that is to say
$$x=-\frac{n\,\,W_{-1}(t)}{(\beta +1) Z \log (n)} \quad \text{where} \qquad t=-\frac{e^{\beta +1} (\beta +1) Z \log (n)}{n}$$ For the worked example, this will give $K=213.69$
For the evaluation of $W(t)$, assuming that $\beta &gt;-1$ (wich makes $t&lt;0$) you can use the series expansion
$$W(t)\approx L_1-L_2+\frac{L_2}{L_1}+\frac{L_2(L_2-2)}{2L_1^2}+\frac{L_2(2L_2^2-9L_2+6)}{6L_1^3}+\cdots$$ where $L_1=\log(-t)$ and $L_2=\log(-L_1)$ .
If you just look for bounds, in the linked page, you will notice that in $2013$ was established the double inequality
$$-1 - \sqrt{2u} - u &lt; W_{-1}\left(-e^{-(u+1)}\right) &lt; -1 - \sqrt{2u} - \tfrac{2}{3}u $$
For the worked example this gives
$208.178 &lt; K &lt;229.578$
K and n are constant integers. $\beta$ is constant real. Being in form a + b Z as a linear relation.. minimum when Z=0.