The next problem, much like my previous problem is to find the values of $m$ $\in$ $\mathbb R^+$ such that the function $f(n)=\displaystyle\sqrt[\displaystyle m]{(n+1)(\ln (n+1) + \ln \ln (n+1))}$ $-$ $\displaystyle\sqrt[\displaystyle m]{n(\ln n + \ln \ln n - 1)}$ $-$ $1$ is decreasing.

In particular, examine the case when $m=2$.

Actually this problem is one of the class of problems that I am investigating right now. The answer to the previous problem has given me the impression that the function must be decreasing for all $n$ greater than some particular $k$. Actually what I want to find that value of $k$ as a function of $m$ if possible. For the excellent proof of my previous problem see Comment on the monotonicity of the function.


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