Let \(\psi _{n}=\left( -1\right) ^{n-1}\psi ^{\left( n\right) }\) for \(n\ge 0\) , where \(\psi ^{\left( n\right) }\) stands for the psi and polygamma functions. For \(p,q\in \mathbb {R}\) and \(\rho =\min \left( p,q\right) \), let

be the divided difference of the functions \(\psi _{n-1}\) for \(x>-\rho \). In this paper, we prove that for \(\left| p-q\right| \gtrless 1\), the sequences \(\{\xi _{n}\left( x\right) \}_{n\in \mathbb {N}}\) and \(\{\eta _{n}\left( x\right) \}_{n\in \mathbb {N}}\) defined by

are strictly decreasing (increasing) and converge to \(\min \left( p,q\right) \). Furthermore, if \(\left| p-q\right| \gtrless 1\), then the functions \(x\mapsto \xi _{n}\left( x\right) \) and \(x\mapsto \eta _{n}\left( x\right) \) are strictly increasing (decreasing) from \(\left( -\rho ,\infty \right) \) onto \(\left( \rho ,\left( p+q-1\right) /2\right) \) (\(\left( \left( p+q-1\right) /2,\rho \right) \)). These not only generalize and strengthen some known results, but also yield several new sharp bounds involving polygamma functions.

让\(\psi _{n}=\left( -1\right) ^{n-1}\psi ^{\left( n\right) }\)为\(n\ge 0\)，其中\ (\psi ^{\left( n\right) }\)代表 psi 和 polygamma 函数。对于\(p,q\in \mathbb {R}\)和\(\rho =\min \left( p,q\right) \)，让

是函数\(\psi _{n-1}\)对于\(x>-\rho \)的除法差。在本文中，我们证明对于\(\left| pq\right| \gtrless 1\)，序列\(\{\xi _{n}\left( x\right) \}_{n\in \ mathbb {N}}\)和\(\{\eta _{n}\left( x\right) \}_{n\in \mathbb {N}}\)定义为

严格减少（增加）并收敛到\(\min \left( p,q\right) \)。此外，如果\(\left| pq\right| \gtrless 1\)，则函数\(x\mapsto \xi _{n}\left( x\right) \)和\(x\mapsto \eta _ {n}\left( x\right) \)从\(\left( -\rho ,\infty \right) \)到\(\left( \rho ,\left( p+q) -1\right) /2\right) \) ( \(\left( \left( p+q-1\right) /2,\rho \right) \) )。这些不仅概括和加强了一些已知结果，而且还产生了几个涉及多伽马函数的新锐界。