Let $\lambda_1 (\cdot)$ be the larger absolute value eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$ the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e. $\lambda_1 (\cdot) \ge \lambda_2 (\cdot)$. Is it true that $$\Big\lambda_1 (A+B)\lambda_1 (A)\Big^{1/3}+\Big\lambda_2 (A+B)\lambda_2 (A)\Big^{1/3}\leq\lambda_1 (B)^{1/3}+\lambda_2 (B)^{1/3}$$ for any two $2\times2$ symmetric real matrices $A$ and $B$? Thanks a lot!
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The alleged inequality is false, even if you restrict $A$ and $B$ to be positive definite matrices. Consider the following, $$ A = \begin{bmatrix} 1.2281 & 0.6361\\\\ 0.6361 & 1.9690 \end{bmatrix},\quad\quad B = \begin{bmatrix} 3.7829 &0.6021\\\\ 0.6021 & 0.4002 \end{bmatrix}. $$ Then, we have the following: \begin{eqnarray*} \lambda(A+B) = (5.0114, 2.3687)\\\\ \lambda(A) = (2.3347, 0.8624)\\\\ \lambda(B) = (3.8868, 0.2962) \end{eqnarray*} From, which we see that \begin{eqnarray*}\ \ \lambda_1(A+B) \lambda_1(A)\ \ ^{1/3} + \ \ \lambda_2(A+B)\lambda_2(A)\ \ ^{1/3} & = & 2.5348\\\\ \lambda_1(B)^{1/3} + \lambda_2(B)^{1/3} &=& 2.2389 \end{eqnarray*} 

