论文标题
关于与几何平均值有关的规范不平等现象
On norm inequalities related to the geometric mean
论文作者
论文摘要
令$ a_i $和$ b_i $为所有$ i = 1，\ cdots，m。$的正确定矩阵，显示出$$ \ left | \ weft | \ sum_ {i = 1}^m（a_i^2 \ sharp） b_i^2）^r \右| \ right | _1 \ leq \ left | \ left | \ left | \ left（\ left（\ sum _ {i = 1}^ma_i \ right）^{\ frac {pr} {pr} {_ _ 2}}}}}}} \ sum_ left（\ sum_左（\ sum_ = 1 = 1 }^mb_i \ right）^{pr} \ left（\ sum_ {i = 1}^ma_i \ right）^{\ frac {rp} {rp} {_ 2}}} \ right）^{\ frac {1} {1} {1} {p}} {p}}}}}} {p}}} \ right | \ right | \ right | \ right | rirt | _1，$ _1，$所有$ p> 0 $，对于所有$ r \ geq1。 我们猜想所有单位不变的规范也是如此。我们对$ m = 2，$ $ $ $ p \ geq1 $，$ r \ geq1 $和所有单位不变规范的情况给出了肯定的答案。换句话说，这表明$$ \左| \左| \ left | \ left（a^{^{^2} \ sharp b^{^2} \ right）^{r}+\左（c^{^2} \ sharp d^{^2} {^2} \ right） \左| \左| \左| \左（\ left（a+c \ right）^{^\ frac {rp} {_ 2}}}} \ left（b+d \ right）^{rp} {rp} \ l eft（a+c \ right）^{^\ frac {rp} {_ 2}}} \ right）^{\ frac {\ frac {1} {_ p}}} \ right | \ right | \ right | \ right | \ right | right |，$ $所有Unitarly不变规范，对于所有$ p \ geq1 $，对于所有$ r \ geq1 $，其中$ a，b，c，d $都是正定矩阵。这给出了$ m = 2 $的Dinh，Ahsani和Tam所提出的猜想。前面的不等式直接导致了Audenaert \ cite {anifp}的最新结果。
Let $A_i$ and $B_i$ be positive definite matrices for all $i=1,\cdots,m.$ It is shown that $$\left|\left|\sum_{i=1}^m(A_i^2\sharp B_i^2)^r\right|\right|_1\leq\left|\left|\left(\left(\sum_{i=1}^mA_i\right)^{\frac{pr}{_2}}\left(\sum_{i=1}^mB_i\right)^{pr}\left(\sum_{i=1}^mA_i\right)^{\frac{rp}{_2}}\right)^{\frac{1}{p}}\right|\right|_1,$$for all $p>0$ and for all $r\geq1.$ We conjecture this inequality is also true for all unitarily invariant norms. We give an affirmative answer to the case of $m=2,$ $p\geq1$, $r\geq1$ and for all unitarily invariant norms. In other words, it is shown that $$\left|\left|\left|\left(A^{^2}\sharp B^{^2}\right)^{r}+\left(C^{^2}\sharp D^{^2}\right)^{r}\right|\right|\right|\leq \left|\left|\left|\left(\left(A+C\right)^{^\frac{rp}{_2}}\left(B+D\right)^{rp}\left(A+C\right)^{^\frac{rp}{_2}}\right)^{\frac{1}{_p}}\right|\right|\right|,$$for all unitarly invariant norms, for all $p\geq1$ and for all $r\geq1$, where $A,B,C,D$ are positive definite matrices. This gives an affirmative answer to the conjecture posed by Dinh, Ahsani and Tam in the case of $m=2$. The preceding inequalities directly lead to a recent result of Audenaert \cite{ANIFP}.
