Optimisation of two parameters. I would like to optimize firstly $D >0$ then $K\in \mathbb{N}$ to get this inequality
$$
\frac{TL}{K^{\beta}} + DT + 3KL + \frac{16K\log(T)}{D} \le C_{L,\beta} T^{\frac{\beta +1}{2\beta+1}} \log(T)^{\frac{\beta}{2\beta+1}} 
$$
with $\beta \in ]0;1]$, $L>0$, $T \in \mathbb{N}-\{0\}$.
I attempt was to say that $T\ge 3$ otherwise it's complicated, maybe we can do better, surely. And then I chosed $D = \log(T)^{\frac{1}{2(2\beta+1)}}LT^{\frac{-1}{2\beta+1}} $ then $K = \left \lfloor \sqrt{T^{1-\frac{1}{2\beta+1}} } \right \rfloor + 1$ so $C_{\beta,L} = 42L$ 
It's weird there's no $\beta$ in $C_{\beta,L}$ ! That's why I'm here, I would like to learn how to deal with those kind of question because my method was really... Wild !
Thanks and regards. 
 A: My following straightforward approach provides a similar bound. 
Denote the left hand side of the inequality by $A$. 
Assume first that $T>1$, so $\log T>0$. Then, by AM-GM inequality, $DT+\frac{16K\log T}{D}$ is minimized when $DT=\frac{16K\log T}{D}$, that is $D=\sqrt{\frac{16K\log T}{T}}$ and the minimum value equals $8\sqrt{KT\log T}$. Now $A$ becomes $\frac{TL}{K^{\beta}} +  3KL +8\sqrt{KT\log T}$. 
The calculation of the exact minimum of this expression with respect to $K$ looks complicated, so we’ll try to obtain possible weaker, but simpler bounds as follows. Remark that
$\frac{TL}{K^{\beta}} +  3KL +8\sqrt{KT\log T}\le (L+1)\left(\frac{T}{K^{\beta}} +  3K +8\sqrt{KT\log T}\right)$.
Asymptotic evaluations suggest to pick $K=\left(\frac{T}{\log T}\right)^{\frac 1{1+2\beta}}$ (this value can be non-integer, but I expect that if we change $K$ to a closest integer value then the expression will not change much). Then  
$\frac{T}{K^{\beta}} +  3K +8\sqrt{KT\log T}=9T^{\frac{\beta+1}{2\beta+1}}(\log T)^ {\frac{\beta}{2\beta+1}}+3T^{\frac 1{1+2\beta}}(\log T)^{-\frac 1{1+2\beta}}$.
If $T=1$ then $A=\frac{L}{K^{\beta}} + D + 3KL$. If we pick $K=1$ and $D=L$ then $A=5L$.
