Post Closed as "no longer relevant" by Andres Caicedo, Andy Putman, S. Sra, Will Jagy, Harald Hanche-Olsen
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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! By the way, this question has been posted on Stack Exchange, but I think MO is a better place to post it. [this question has beeen answered by Suvrit]

A further question that is more interesting is: Is it true that $$\left|\left|\lambda_{1}\left(A+B\right)\right|^{1/3}-\left|\lambda_{1}\left(A\right)\right|^{1/3}\right|+\left|\left|\lambda_{2}\left(A+B\right)\right|^{1/3}-\left|\lambda_{2}\left(A\right)\right|^{1/3}\right|\leq\left|\lambda_{1}\left(B\right)\right|^{1/3}+\left|\lambda_{2}\left(B\right)\right|^{1/3}$$ for any $2\times2$ symmetric real matrices $A$ and $2\times2$ diagonal real matrices $B$? Thanks a lot for any helpful answers!

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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! By the way, this question has been posted on Stack Exchange, but I think MO is a better place to post it. [this question has beeen answered by Suvrit]

A further question that is more interesting is: Is it true that $$\left|\left|\lambda_{1}\left(A+B\right)\right|^{1/3}-\left|\lambda_{1}\left(A\right)\right|^{1/3}\right|+\left|\left|\lambda_{2}\left(A+B\right)\right|^{1/3}-\left|\lambda_{2}\left(A\right)\right|^{1/3}\right|\leq\left|\lambda_{1}\left(B\right)\right|^{1/3}+\left|\lambda_{2}\left(B\right)\right|^{1/3}$$ for any $2\times2$ symmetric real matrices $A$ and $2\times2$ diagonal real matrices $B$? Thanks a lot for any helpful answers!

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