# Some Refinements of Selberg Inequality and Related Results

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## Abstract

**:**

## 1. Introduction

## 2. Generalized Selberg Inequality

**Proposition 1.**

**Proof.**

**Definition 1.**

**Remark 1.**

- (1)
- Utilizing the Selberg operator, we can rephrase the statement (SI) as follows:$$0\le \langle {S}_{\mathcal{Z}}u,u\rangle =\sum _{p=1}^{n}\frac{|\langle u,{z}_{p}\rangle {|}^{2}}{{\sum}_{q=1}^{n}\left|\langle {z}_{p},{z}_{q}\rangle \right|}\le \langle u,u\rangle ,$$for any $u\in \mathrm{E}$. As a consequence, we can conclude that all Selberg operators are positive contractions, denoting that $0\le {S}_{\mathcal{Z}}\le I$. Moreover, this operator inequality allows us to infer the following:$$0\le I-{S}_{\mathcal{Z}}\le I.$$
- (2)
- It follows from (8) that$$\omega (I-{S}_{\mathcal{Z}})=\parallel I-{S}_{\mathcal{Z}}\parallel \le 1.$$

**Theorem 1.**

**Proof.**

**Proposition 2.**

**Proof.**

**Corollary 1.**

**Corollary 2.**

**Proof.**

**Proposition 3.**

**Proof.**

**Lemma 1.**

- (1)
- $\parallel {S}_{\mathcal{Z}}u\parallel =\parallel u\parallel $.
- (2)
- $\langle {S}_{\mathcal{Z}}u,u\rangle ={\parallel u\parallel}^{2}.$
- (3)
- $u\in \mathcal{N}(I-{S}_{\mathcal{Z}})$.

**Proof.**

**Lemma 2.**

- (1)
- $\parallel (I-{S}_{\mathcal{Z}})u\parallel =\parallel u\parallel $.
- (2)
- $u\in \mathcal{N}\left({S}_{\mathcal{Z}}\right)$.
- (3)
- $\langle u,{z}_{p}\rangle =0$ for all $p=1,\dots ,n.$

**Proof.**

**Theorem 2.**

**Proof.**

**Theorem 3.**

**Proof.**

**Proposition 4.**

**Proof.**

**Proposition 5.**

**Proof.**

**Remark 2.**

**Theorem 4.**

**Proof.**

**Remark 3.**

- (1)
- From the the previous statement, we have$$\begin{array}{c}\hfill \left(\right)open="|"\; close="|">\langle {S}_{\mathcal{Z}}u,v\rangle -\frac{1}{2}\langle u,v\rangle \\ \le & \frac{1}{2}\parallel u\parallel \parallel v\parallel .\hfill \end{array}$$If we consider $\mathcal{Z}=\left\{z\right\}$ with $z\ne 0$, then ${S}_{\mathcal{Z}}=\frac{z\otimes z}{{\parallel z\parallel}^{2}}$ is an orthogonal projection onto the subspace spanned by $\left\{z\right\}$. Consequently, we obtain the well-known Richard’s inequality (see [23]):$$\left(\right)open="|"\; close="|">\langle u,z\rangle \langle z,v\rangle -\frac{1}{2}\langle u,v\rangle {\parallel z\parallel}^{2}$$
- (2)
- Using the fact that $\parallel 2(I-{S}_{\mathcal{Z}})-I\parallel =\parallel I-2{S}_{\mathcal{Z}}\parallel \le 1$, and applying similar ideas used in the proof of Theorem 4, we can establish that for any $u,v\in \mathrm{E}$,$$\begin{array}{cc}\hfill |\langle (I-{S}_{\mathcal{Z}})u,v\rangle |\phantom{\rule{1.em}{0ex}}& \le \left(\right)open="|"\; close="|">\langle (I-{S}_{\mathcal{Z}})u,v\rangle -\frac{1}{2}\langle u,v\rangle +\frac{1}{2}\left|\langle u,v\rangle \right|\hfill \end{array}$$

**Corollary 3.**

**Theorem 5.**

**Proof.**

**Corollary 4.**

**Proof.**

**Theorem 6.**

**Proof.**

**Remark 4.**

- (1)
- In the work of Lin [26], the investigation of covariance-variance for bounded linear operators defined on a Hilbert space $\mathrm{E}$ was initiated. Let us recall some definitions introduced in that article. Let $R,T\in \mathit{L}\left(\mathrm{E}\right)$ and $z\ne 0$. The covariance of R and T is a mapping $Co{v}_{z}(R,T):\mathrm{E}\to \mathbb{C}$ defined by$$Co{v}_{z}(R,T)u={\parallel z\parallel}^{2}\langle Ru,Tu\rangle -\langle Ru,z\rangle \langle z,Tu\rangle .$$If $R=T$ we obtain the variance of S$$Va{r}_{z}\left(R\right)u=Co{v}_{z}(R,R)u={\parallel z\parallel}^{2}{\parallel Ru\parallel}^{2}-{\left|\langle Ru,z\rangle \right|}^{2}.$$In particular, if in the first inequality of (24) we consider $\mathcal{Z}=\left(\right)open="\{"\; close="\}">\frac{z}{\parallel z\parallel}$ and we replace u and v by $Ru$ and $Tu$, respectively, then$$\begin{array}{ccc}\hfill |Co{v}_{z}{(R,T)u|}^{2}& =& {\parallel z\parallel}^{2}{|\langle {S}_{\mathcal{Z}}\left(Ru\right),Tu\rangle -\langle Ru,Tu\rangle |}^{2}\hfill \\ & \le & {\parallel z\parallel}^{2}{(\parallel Ru\parallel}^{2}-\langle {S}_{\mathcal{Z}}\left(Ru\right),Ru\rangle {\left)\right(\parallel Tu\parallel}^{2}-\langle {S}_{\mathcal{Z}}\left(Tu\right),Tu\rangle )\hfill \\ & =& \left(Va{r}_{z}\left(R\right)u\right)\left(Va{r}_{z}\left(T\right)u\right).\hfill \end{array}$$
- (2)
- By utilizing the second inequality of (24) and (SI), we can provide an alternative proof that the Selberg operator ${S}_{\mathcal{Z}}$ satisfies Buzano’s inequality (refer to Theorem 4). Specifically, we have:$$|\langle {S}_{\mathcal{Z}}u,v\rangle -\langle u,v\rangle |\le \parallel u\parallel \parallel v\parallel -{\langle {S}_{\mathcal{Z}}u,u\rangle}^{\frac{1}{2}}{\langle {S}_{\mathcal{Z}}v,v\rangle}^{\frac{1}{2}},$$for any $u,v\in \mathrm{E}.$ As a consequence of (5), we have$$|\langle {S}_{\mathcal{Z}}u,v\rangle -\langle u,v\rangle |\le \parallel u\parallel \parallel v\parallel -|\langle {S}_{\mathcal{Z}}u,v\rangle |.$$

**Theorem 7.**

- (1)
- Bessel inequality—If $\mathcal{E}=\{{e}_{i}:i=1,\dots ,n\}$ is an orthonormal set in $\mathrm{E}$, then$$\sum _{i=1}^{n}|\langle u,{e}_{i}\rangle {|}^{2}\le {\parallel u\parallel}^{2},$$for any $u\in \mathrm{E}$.
- (2)
- Cauchy–Schwarz inequality—For any $u,v\in \mathrm{E}$, we have$$|\langle u,v\rangle |\le \parallel u\parallel \parallel v\parallel .$$
- (3)
- Selberg inequality—For given nonzero vectors $\mathcal{Z}=\{{z}_{p}:p=1,\dots ,n\}\subseteq \mathrm{E},$ the inequality$$\sum _{p=1}^{n}\frac{|\langle u,{z}_{p}\rangle {|}^{2}}{{\sum}_{q=1}^{n}\left|\langle {z}_{p},{z}_{q}\rangle \right|}\le {\parallel u\parallel}^{2},$$holds for all $u\in \mathrm{E}.$

**Proof.**

**Remark 5.**

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Altwaijry, N.; Conde, C.; Dragomir, S.S.; Feki, K.
Some Refinements of Selberg Inequality and Related Results. *Symmetry* **2023**, *15*, 1486.
https://doi.org/10.3390/sym15081486

**AMA Style**

Altwaijry N, Conde C, Dragomir SS, Feki K.
Some Refinements of Selberg Inequality and Related Results. *Symmetry*. 2023; 15(8):1486.
https://doi.org/10.3390/sym15081486

**Chicago/Turabian Style**

Altwaijry, Najla, Cristian Conde, Silvestru Sever Dragomir, and Kais Feki.
2023. "Some Refinements of Selberg Inequality and Related Results" *Symmetry* 15, no. 8: 1486.
https://doi.org/10.3390/sym15081486