Infimum and bound of the inequality.

Question

In a paper, I got the following inequality but can't understand why it's true. Any reference or explanation of the proof is welcomed.

$$\text{inf}_{\eta\in (0,1)}\tfrac{1}{\eta^d (1-\eta)^k}= \tfrac{(d+k)^{(d+k)}}{k^k d^d}\leq \frac{e^k}{k^k}(d+k)^k $$ I guess $d,k$ are constant and $e$ denote exponential. The paper doesn't mentioned it.

Answer 1

I will assume that $d,k>0$.

The first inequality follows by taking $\eta =\frac 1 {1+\frac k d}$ and the second inequality follows from the fact that $(1+\frac k d)^{d} \leq e^{k}$.