In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$.
Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are
$$U_n:=U_n(x)=\sigma_n(x)/n^{x+1} \quad \text{and}\quad L_n:= L_n(x)=\sigma_{n-1}(x)/n^{x+1}.$$
He proved (by mathematical induction) that $U_n>U_{n+1}$ and $L_n<L_{n+1}$.
A later proof of $U_n>U_{n+1}$ was obtained by showing that the function
$$h(x):= h_n(x)=(\sigma_{n+1}/\sigma_n(x))(n/(n+1))^x$$
is strictly decreasing on all of $\mathbf{R}$.
Soon afterwards he came to realize that the monotonicity of $h(x)$ would be a consequence of the logarithmic convexity of $Q(x):=Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ on all of $\mathbf{R}$.
Various numerical tests on $Q(x)$ were performed, but no proof was found.
Another application: From the logarithmic convexity of $Q(x)$ one may also obtain a simple proof of a conjecture made by H. Alzer and A. A. Jagers: $$f(x):=f_n(x)=\Bigl(\frac{\frac{1}{n+1}\sigma_{n+1}(x)}{\frac1n \sigma_n(x)}\Bigr)^{1/x}$$ is strictly increasing for $x>0$.
Question: Is for every (fixed) integer $n\ge2$, the function
$$x\in\mathbf{R}\mapsto Q_n(x):=(1^x+2^x+\cdots+n^x+(n+1)^x)/(1^x+2^x+\cdots+n^x)$$ logarithmically convex on all of $\mathbf{R}$?
