I need your expertise in solving the following inequality:
Given $n \in \mathbb{Z}^+$, $a \in [0,\infty)$ and $b \in \mathbb{R}^+$ such that $ n > b \geq a$ ($b$ cant be zero).
Can the following be proven:
$$ \frac{1+a}{n - b} \leq \log(n)^c \cdot \left( \frac{1}{n} + \frac{a}{b} \right), $$ where $c \in \mathcal{O}(1)$ (i.e. c can be thought as a constant).
Please advise and thanks in advance.
The answer is negative. Indeed, assume that such $c$ exists. Since both $\frac {1+a}{n-b}$ and $\frac 1n+\frac ab$ are positive, $d=(\log n)^c$ should be positive. We show that $d$ should be zero, obtaining a contadiction. We have
$$d\le \left(\frac 1n+\frac ab\right)\frac {n-b}{1+a}=\frac {na}{b(1+a)}+\frac{1-a}{1+a}-\frac{b}{n(1+a)},$$
and this inequality holds for any admissible $b$. Since when $b$ is growing to $n$ both $\frac {na}{b(1+a)}$ and $-\frac{b}{n(1+a)}$ are diminishing, the inequality holds for each $b$ iff it holds for $b=n$ that is iff
$$d\le \frac {ba}{b(1+a)}+\frac{1-a}{1+a}-\frac{b}{b(1+a)}=0.$$