In this paper, we discuss new inequalities for accretive matrices through non-standard domains. In particular, we present several relations for $$A^r$$ and $$A\sharp _rB$$ , when A, B are accretive and $$r\in (-1,0)\cup (1,2).$$ This complements the well-established discussion of such quantities for accretive matrices when $$r\in [0,1],$$ and provides accretive versions of known results for positive matrices. Among many other results, we show that the accretive matrices A, B satisfy $$\begin{aligned} \mathfrak {R}(A\sharp _rB)\le \mathfrak {R}A\sharp _r \mathfrak {R}B, r\in (-1,0)\cup (1,2). \end{aligned}$$ This, and other results, gain their significance due to the fact that they are reversed when $$r\in (0,1).$$

中文翻译：

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关于增生矩阵的加权几何平均

在本文中，我们通过非标准域讨论了增积矩阵的新不等式。特别地，我们给出了 $$A^r$$ 和 $$A\sharp _rB$$ 的几个关系，当 A, B 是增生的并且 $$r\in (-1,0)\cup (1,2) .$$ 当 $$r\in [0,1],$$ 时，这补充了对增加矩阵的此类数量的完善讨论，并提供了正矩阵已知结果的增加版本。在许多其他结果中，我们表明增生矩阵 A、B 满足 $$\begin{aligned} \mathfrak {R}(A\sharp _rB)\le \mathfrak {R}A\sharp _r \mathfrak {R}B , r\in (-1,0)\cup (1,2)。\end{aligned}$$ 这个结果和其他结果之所以重要，是因为当 $$r\in (0,1).$$