In this paper, we present some unitarily invariant norm inequalities for sector matrices involving a special class of functions. In particular, if is a $2n\times 2n$ matrix such that numerical range of Z is contained in a sector region $S_{\alpha } $ for some $\alpha \in [0,\frac{\pi }{2} ) $, then, for a submultiplicative function h of the class $\mathcal{C} $ and every unitarily invariant norm, we have $$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert &\leq \bigl\Vert h^{r} \bigl( \sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{ \frac{1}{s} }, \end{aligned}$$ where r and s are positive real numbers with $\frac{1}{r}+\frac{1}{s}=1 $ and $i,j=1,2$. We also extend some unitarily invariant norm inequalities for sector matrices.

中文翻译：

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范式不等式涉及部门矩阵的一类特殊功能

在本文中，我们提出了涉及一类特殊功能的扇形矩阵的一些单位不变范数不等式。尤其是，如果是一个$ 2n×2n $矩阵，使得Z的数值范围包含在[0，\ frac {\ pi} {2 }）$，然后，对于$ \ mathcal {C} $类的子乘函数h和每个unit不变范数，我们有$$ \ begin {aligned} \ bigl \ Vert h \ bigl（\ vert Z_ {ij} \ vert ^ {2} \ bigr）\ bigr \ Vert＆\ leq \ bigl \ Vert h ^ {r} \ bigl（\ sec（\ alpha）\ vert Z_ {11} \ vert \ bigr）\ bigr \ Vert ^ {\ frac {1} {r}} \ bigl \ Vert h ^ {s} \ bigl（\ sec（\ alpha）\ vert Z_ {22} \ vert \ bigr）\ bigr \ Vert ^ {\ frac {1} {s}}，\ end {aligned} $$，其中r和s是带有$ \ frac {1} {r} + \ frac {1} {s} = 1 $和$ i，j = 1的正实数。 2 $。