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Norm Inequalities via Convex and Log-Convex Functions

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Abstract

In this paper, we study the norm and skew angular distances in a normed space \({\mathscr {X}}\), where convex functions are used to obtain refinements and reverses of some outstanding results in the literature. For example, in this regard, we show that if \(a,b\in {\mathscr {X}}\) are non-zero and if \(p,q>0\) are such that \(\frac{1}{p}+\frac{1}{q}=1\), then

$$\begin{aligned} \begin{aligned} 2\lambda \left( \frac{{{p}^{r}}{{\left\| a \right\| }^{r}}+{{q}^{r}}{{\left\| b \right\| }^{r}}}{2}-{{\left\| \frac{pa+qb}{2} \right\| }^{r}} \right)&\le {{p}^{r-1}}{{\left\| a \right\| }^{r}}+{{q}^{r-1}}{{\left\| b \right\| }^{r}}-{{\left\| a+b \right\| }^{r}} \\&\le 2\mu \left( \frac{{{p}^{r}}{{\left\| a \right\| }^{r}}+{{q}^{r}}{{\left\| b \right\| }^{r}}}{2}-{{\left\| \frac{pa+qb}{2} \right\| }^{r}} \right) , \end{aligned} \end{aligned}$$

where \(r\ge 1\), \(\lambda =\min \left\{ {1}/{p},{1}/{q}\right\} \) and \(\mu =\max \left\{ {1}/{p},{1}/{q} \right\} \). Then we explain how this result extends some known results in the literature. Many other related results will be also shown. Then, with the theme of convexity, we employ a log-convex approach on certain matrix functions to obtain improvements and new sights of some matrix inequalities, including possible bounds of \(\Vert A^{t}XB^{1-t}\Vert ,\) where AB are positive definite matrices, X is an arbitrary matrix, \(\Vert \cdot \Vert \) is a unitarily invariant norm and \(0\le t\le 1.\) Many other results involving matrix and scalar log-convex functions will be presented too.

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The authors would like to express their deep gratitude to the reviewer, for valuable comments.

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Conde, C., Minculete, N., Moradi, H.R. et al. Norm Inequalities via Convex and Log-Convex Functions. Mediterr. J. Math. 20, 6 (2023). https://doi.org/10.1007/s00009-022-02214-z

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  • DOI: https://doi.org/10.1007/s00009-022-02214-z

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