# A few questions regarding the function $f(x) = x+\exp(x)\cdot \log(x)$

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 $$f$$ ?

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 $$x\ge 1$$.