any one can prove following inequality?
That inequality is not defined for $\alpha = 0, 1, 2$. For $\alpha &gt; 2$ you have:
$$
\frac{1}{\alpha-2}  - \frac{1}{\alpha-1} -  \frac{1}{\alpha} =
\frac{\alpha(\alpha-1)-\alpha(\alpha-2)-(\alpha-1)(\alpha-2)}{\alpha(\alpha-1)(\alpha-2)} 
$$
Clearly the denominator is always positive. Regarding the numerator:
$$
\alpha(\alpha-1)-\alpha(\alpha-2)-(\alpha-1)(\alpha-2) =
\alpha(\alpha-1-\alpha+2) - (\alpha-1)(\alpha-2) =\\
\alpha - (\alpha-1)(\alpha-2) =
\alpha - \alpha^2 + 2\alpha  + \alpha  -2=
-\alpha^2 + 4\alpha - 2,
$$
which is negative as soon as $\alpha \ge 4$. In fact, your inequality is only true for $\alpha=3$.