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Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?
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
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