The function $f(x) = x+\exp(x)\log(x)$ occurs prominently at Lagarias inequality: $\sigma(n) \le H_n + \exp(H_n)\log(H_n)$ where $\sigma(n)$ is the sum of divisors, and $H_n$ is the n-th harmonic number.

Let $\tau(n)$ be the number of divisors of $n$, and $(n,k)$ be the gcd of $n,k$. I have been able to prove that the upper bound on the number of divisors: $\tau(n) \le 1/n \sum_{1\le k \le n} {H_{(n,k)} + \exp(H_{(n,k)}) \log(H_{(n,k)})}$ is equivalent to Riemann hypothesis, similar to the Lagarias inequality. Here are a few questions:

1) Is this function $f$ convex? (If so, how to prove it) (I want to apply Jensens inequality)

2) Is there a series expansion for $f$ in terms of $x$ (You can assume $x>1$)

3) Is there a series expansion for the inverse function $f^{-1}$ of $f$ ?

Thanks for your help.


Hint for the convexity part: Note that $$f''(x) = e^x \left(\log x + \frac{2}{x} - \frac{1}{x^2}\right) = e^x \cdot \frac{x^2\log x + 2x - 1}{x^2}.$$

What happens to $x^2\log x + 2x - 1$ as $x\to 0^{+}$, and hence what is the sign of $f''(x)$ for small $x$? And if you only care about $x\ge 1$, consider the sign of this expression when $x\ge 1$.

  • $\begingroup$ thanks. I missed this observation. $\endgroup$ – orgesleka Mar 14 at 11:07

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