An extension of an inequality for ratios of gamma functions

In this paper, we prove that for $x+y>0$ and $y+1>0$ the inequality {equation*} \frac{[\Gamma(x+y+1)/\Gamma(y+1)]^{1/x}}{[\Gamma(x+y+2)/\Gamma(y+1)]^{1/(x+1)}} 1$and reversed if$x<1$and that the power$\frac12$is the best possible, where$\Gamma(x)$ is the Euler gamma function. This extends the result in [Y. Yu, \textit{An inequality for ratios of gamma functions}, J. Math. Anal. Appl. \textbf{352} (2009), no.~2, 967\nobreakdash--970.] and resolves an open problem posed in [B.-N. Guo and F. Qi, \emph{Inequalities and monotonicity for the ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2, 239\nobreakdash--247.].Comment: 8 page

One-step implementation of multi-qubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity

The diamond nitrogen-vacancy (NV) center is an excellent candidate for quantum information processing, whereas entangling separate NV centers is still of great experimental challenge. We propose an one-step conditional phase flip with three NV centers coupled to a whispering-gallery mode cavity by virtue of the Raman transition and smart qubit encoding. As decoherence is much suppressed, our scheme could work for more qubits. The experimental feasibility is justified.Comment: 3 pages, 2 figures, Accepted by Appl. Phys. Let