Correction to: Int J Theor Phys (2022) 61:185

https://doi.org/10.1007/s10773-022-05160-4

The original version of this article unfortunately contained two mistakes.

  1. (1)

    The second author Zhaoqi Wu is not connected to affiliation 2. Instead, Zhaoqi Wu is connected to affiliation 1. The third author Shao-Ming Fei belongs to both affiliation 2 and affiliation 3, instead of affiliation 3. The correct information is shown in this erratum.

  2. (2)

    The authors wish to correct a typographical error found in the original article. Equation (32) and the equation in the proof of Theorem 8 has writing mistake. The correct equation (32) is found below:

    $$\begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{N}\mathrm{K}_{\rho,\gamma}^{\alpha,\upbeta}(A_{i}) &\!\geq\!\! & \left(\begin{array}{c} N-2 \\ k-1 \end{array} \right)^{-1} \left[\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq N}\mathrm{K}_{\rho,\gamma}^{\alpha,\upbeta}\left(\sum\limits_{j=1}^{k}A_{i_{j}}\right) - \left(\begin{array}{c} N-2 \\ k-2 \end{array} \right) \left(\begin{array}{c} N - 1 \\ k-1 \end{array} \right)^{-2} \right. \\ &&\left. \left(\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\upbeta}\left(\sum\limits_{j=1}^{k}A_{i_{j}}\right)}\right)^{2} \right]~,\alpha,\upbeta\geq 0,\alpha+\upbeta\leq 1,0\leq \gamma \leq 1.\\ \end{array}$$
    (32)

    instead of

    $$\begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{N}\mathrm{K}_{\rho,\gamma}^{\alpha,\upbeta}(A_{i}) &\geq & \left(\begin{array}{c} N-2 \\ k-1 \end{array} \right)^{-1} \left[\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq N}\mathrm{K}_{\rho,\gamma}^{\alpha,\upbeta}\left(\sum\limits_{j=1}^{k}A_{i_{j}}\right)-\left(\right) \left(\begin{array}{c} N-1 \\ k-1 \end{array} \right)^{-2} \right. \\ &&\left. \left(\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\upbeta}\left(\sum\limits_{j=1}^{k}A_{i_{j}}\right)}\right)^{2} \right]~,\alpha,\upbeta\geq 0,\alpha+\upbeta\leq 1,0\leq \gamma \leq 1.\\ \end{array}$$
    (32)

In the proof of Theorem 8, the correct equation is found below:

$$\begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{N}\|u_{i}\|^{2} &\geq & {\left(\begin{array}{c} N-2 \\ k-1 \end{array} \right)}^{-1}\left[\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq N}\|u_{i_{1}}+\cdots+u_{i_{k}}\|^{2}-\left(\begin{array}{c} N-2 \\ k-2 \end{array} \right) \left(\begin{array}{c} N-1 \\ k-1 \end{array} \right)^{-2} \right. \\ &&\left.\left(\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq N} {\|u_{i_{1}}+\cdots+u_{i_{k}}\|}\right)^{2}\right]. \end{array}$$

instead of

$$\begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{N}\|u_{i}\|^{2} &\geq & {\left(\begin{array}{c} N-2 \\ k-1 \end{array} \right)}^{-1}\left[\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq N}\|u_{i_{1}}+\cdots+u_{i_{k}}\|^{2}-\left(\right) \left(\begin{array}{c} N-1 \\ k-1 \end{array} \right)^{-2} \right. \\ &&\left.\left(\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq N} {\|u_{i_{1}}+\cdots+u_{i_{k}}\|}\right)^{2}\right]. \end{array}$$