# A Brief Overview and Survey of the Scientific Work by Feng Qi

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## Abstract

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## 1. Introduction

## 2. Concrete Contributions

#### 2.1. Bell Numbers and Inequalities

- 1.
- Let $\mathit{a}=({a}_{1},{a}_{2},\cdots ,{a}_{n})$ and $\mathit{b}=({b}_{1},{b}_{2},\cdots ,{b}_{n})$ be two non-increasing tuples of non-negative integers such that ${\sum}_{i=1}^{k}{a}_{i}\ge {\sum}_{i=1}^{k}{b}_{i}$ for $1\le k\le n-1$ and ${\sum}_{i=1}^{n}{a}_{i}={\sum}_{i=1}^{n}{b}_{i}$. Then$${B}_{{a}_{1}}{B}_{{a}_{2}}\cdots {B}_{{a}_{n}}\ge {B}_{{b}_{1}}{B}_{{b}_{2}}\cdots {B}_{{b}_{n}}.$$
- 2.
- If $\ell \ge 0$ and $n\ge k\ge 0$, then we have$${B}_{n+\ell}^{k}{B}_{\ell}^{n-k}\ge {B}_{k+\ell}^{n}.$$
- 3.
- If $\ell \ge 0$, $n\ge k\ge m$, $2k\ge n$, and $2m\ge n$, then we have$${B}_{k+\ell}{B}_{n-k+\ell}\ge {B}_{m+\ell}{B}_{n-m+\ell}.$$
- 4.
- If $k\ge 0$ and $n\in \mathbb{N}$, then we have$${\left(\prod _{\ell =0}^{n}{B}_{k+2\ell}\right)}^{1/(n+1)}\ge {\left(\prod _{\ell =0}^{n-1}{B}_{k+2\ell +1}\right)}^{1/n}.$$

#### 2.2. Partial Bell Polynomials

- 1.
- (a)
- For $m\in \mathbb{N}$ and $\left|t\right|<1$, the function ${\left(\frac{arcsint}{t}\right)}^{m}$, whose value at $t=0$ is defined to be 1, has Maclaurin’s series expansion$${\left(\frac{arcsint}{t}\right)}^{m}=1+\sum _{k=1}^{\infty}{(-1)}^{k}\frac{Q(m,2k;2)}{\left(\genfrac{}{}{0pt}{}{m+2k}{m}\right)}\frac{{\left(2t\right)}^{2k}}{\left(2k\right)!},$$$$Q(m,k;\alpha )=\sum _{\ell =0}^{k}\left(\genfrac{}{}{0pt}{}{m+\ell -1}{m-1}\right)s(m+k-1,m+\ell -1){\left(\frac{m+k-\alpha}{2}\right)}^{\ell}$$$$\frac{{[ln(1+x)]}^{k}}{k!}=\sum _{n=k}^{\infty}s(n,k)\frac{{x}^{n}}{n!},\phantom{\rule{1.em}{0ex}}\left|x\right|<1.$$
- (b)
- For $k,n\ge 0$ and ${x}_{m}\in \mathbb{C}$ with $m\in \mathbb{N}$, we have$${B}_{2n+1,k}\left(0,{x}_{2},0,{x}_{4},\cdots ,\frac{1+{(-1)}^{k}}{2}{x}_{2n-k+2}\right)=0.$$For $k,n\in \mathbb{N}$ such that $2n\ge k\in \mathbb{N}$, we have$$\begin{array}{c}{B}_{2n,k}\left(0,\frac{1}{3},0,\frac{9}{5},0,\frac{225}{7},\cdots ,\frac{1+{(-1)}^{k+1}}{2}\frac{{[(2n-k)!!]}^{2}}{2n-k+2}\right)\hfill \\ \hfill ={(-1)}^{n+k}\frac{\left(4n\right)!!}{(2n+k)!}\sum _{q=1}^{k}{(-1)}^{q}\left(\genfrac{}{}{0pt}{}{2n+k}{k-q}\right)Q(q,2n;2),\end{array}$$

Maclaurin’s series expansion (1) was recovered in (Section 6 [9]) and was generalized in (Section 4 [10]) as$${\left(\frac{arcsint}{t}\right)}^{\alpha}=1+\sum _{n=1}^{\infty}{(-1)}^{n}\left[\sum _{k=1}^{2n}\frac{{(-\alpha )}_{k}}{(2n+k)!}\sum _{q=1}^{k}{(-1)}^{q}\left(\genfrac{}{}{0pt}{}{2n+k}{k-q}\right)Q(q,2n;2)\right]{\left(2t\right)}^{2n}$$$${\left(\alpha \right)}_{m}=\prod _{k=0}^{m-1}(\alpha +k)=\left\{\begin{array}{cc}\alpha (\alpha +1)\cdots (\alpha +m-1),\hfill & m\in \mathbb{N};\hfill \\ 1,\hfill & m=0.\hfill \end{array}\right.$$- 2.
- In [9], among other things, by establishing the Taylor series expansion$${\left[\frac{{(arccosx)}^{2}}{2(1-x)}\right]}^{k}=1+\left(2k\right)!\sum _{n=1}^{\infty}\frac{Q(2k,2n;2)}{(2k+2n)!}{\left[2(x-1)\right]}^{n}$$$$\begin{array}{c}{B}_{m,k}\left(-\frac{1}{12},\frac{2}{45},-\frac{3}{70},\frac{32}{525},-\frac{80}{693},\cdots ,\frac{(2m-2k+2)!!}{(2m-2k+4)!}Q(2,2m-2k+2;2)\right)\hfill \\ \hfill ={(-1)}^{k}\left[2(m-k)\right]!!\left(\genfrac{}{}{0pt}{}{m}{k}\right)\sum _{j=1}^{k}{(-1)}^{j}\left(2j\right)!\left(\genfrac{}{}{0pt}{}{k}{j}\right)\frac{Q(2j,2m;2)}{(2j+2m)!}\end{array}$$$${\left[\frac{{(arccosx)}^{2}}{2(1-x)}\right]}^{\alpha}=1+\sum _{n=1}^{\infty}\left[\sum _{j=1}^{n}\frac{{(-\alpha )}_{j}}{j!}\sum _{\ell =1}^{j}{(-1)}^{\ell}(2\ell )!\left(\genfrac{}{}{0pt}{}{j}{\ell}\right)\frac{Q(2\ell ,2n;2)}{(2\ell +2n)!}\right]{\left[2(x-1)\right]}^{n}$$
- 3.
- In [10], among other things, by establishing the specific values$${B}_{2r+k,k}\left(1,0,1,0,9,0,225,0,\cdots ,{[(2r-3)!!]}^{2},0,{[(2r-1)!!]}^{2}\right)={(-1)}^{r}{2}^{2r}Q(k,2r;2)$$$${B}_{2r+k-1,k}\left(1,0,1,0,9,0,225,0,\cdots ,{[(2r-3)!!]}^{2},0\right)=0$$$$\begin{array}{c}{\left(\frac{2arccost}{\pi}\right)}^{\alpha}=1+\sum _{r=1}^{\infty}{(-1)}^{r}\left[\sum _{\ell =1}^{r}{(-1)}^{\ell}\frac{{(-\alpha )}_{2\ell -1}}{{\pi}^{2\ell -1}}Q(2\ell -1,2r-2\ell ;2)\right]\frac{{\left(2t\right)}^{2r-1}}{(2r-1)!}\hfill \\ \hfill +\frac{{(-\alpha )}_{2}}{{\pi}^{2}}\frac{{\left(2t\right)}^{2}}{2!}+\sum _{r=2}^{\infty}{(-1)}^{r}\left[\sum _{\ell =1}^{r}{(-1)}^{\ell}\frac{{(-\alpha )}_{2\ell}}{{\pi}^{2\ell}}Q(2\ell ,2r-2\ell ;2)\right]\frac{{\left(2t\right)}^{2r}}{\left(2r\right)!}\end{array}$$
- 4.
- In [11], among other things, by establishing a special case of (3) and the explicit formula$$\begin{array}{c}{B}_{2m,k}\left(0,-\frac{1}{3},0,\frac{1}{5},\cdots ,\frac{{(-1)}^{m}}{2m-k+2}sin\frac{k\pi}{2}\right)\hfill \\ \hfill ={(-1)}^{m+k}\frac{{2}^{2m}}{k!}\sum _{j=1}^{k}{(-1)}^{j}\left(\genfrac{}{}{0pt}{}{k}{j}\right)\frac{T(2m+j,j)}{\left(\genfrac{}{}{0pt}{}{2m+j}{j}\right)},\phantom{\rule{1.em}{0ex}}2m\ge k\ge 1,\end{array}$$Qi showed that,
- (a)
- when $\alpha \ge 0$, the series expansions$${sinc}^{\alpha}z=1+\sum _{q=1}^{\infty}{(-1)}^{q}\left[\sum _{k=1}^{2q}\frac{{(-\alpha )}_{k}}{k!}\sum _{j=1}^{k}{(-1)}^{j}\left(\genfrac{}{}{0pt}{}{k}{j}\right)\frac{T(2q+j,j)}{\left(\genfrac{}{}{0pt}{}{2q+j}{j}\right)}\right]\frac{{\left(2z\right)}^{2q}}{\left(2q\right)!}$$
- (b)

where$$sincz=\{\begin{array}{cc}\frac{sinz}{z},\hfill & z\ne 0\\ 1,\hfill & z=0\end{array}$$$$T(n,\ell )=\{\begin{array}{cc}1,\hfill & (n,\ell )=(0,0)\\ 0,\hfill & n\in \mathbb{N},\ell =0\\ \frac{1}{\ell !}\sum _{j=0}^{\ell}{(-1)}^{j}\left(\genfrac{}{}{0pt}{}{\ell}{j}\right){\left(\frac{\ell}{2}-j\right)}^{n},\hfill & n,\ell \in \mathbb{N}\end{array}$$

#### 2.3. Wallis Ratio

#### 2.4. Additivity of Polygamma Functions

#### 2.5. Bounds for Mathematical Means in Terms of Mathematical Means

#### 2.6. Complete Elliptic Integrals

#### 2.7. Matrices

- 1.
- Suppose that $A,B,H,E\in {\mathbb{C}}^{n\times n}$ are Hermitian complex matrices of format $n\times n$, that B is positive definite, that $\nu ={\parallel E\parallel}_{2}/{\lambda}_{n}\left(B\right)<1$, and that the positive integers $i,j,k,\ell ,m\in \mathbb{N}$ satisfy $j+k-1\le i\le \ell +m-n-1$.
- (a)
- If ${\lambda}_{i}(A+H)\ge 0$, then$$\frac{{\lambda}_{\ell}\left(A{B}^{-1}\right)+{\lambda}_{m}\left(H{B}^{-1}\right)}{1+\nu}\le {\lambda}_{i}\left((A+H){(B+H)}^{-1}\right)\le \frac{{\lambda}_{j}\left(A{B}^{-1}\right)+{\lambda}_{k}\left(H{B}^{-1}\right)}{1-\nu}.$$
- (b)
- If ${\lambda}_{i}(A+H)\le 0$, then$$\frac{{\lambda}_{j}\left(A{B}^{-1}\right)+{\lambda}_{k}\left(H{B}^{-1}\right)}{1-\nu}\le {\lambda}_{i}\left((A+H){(B+H)}^{-1}\right)\le \frac{{\lambda}_{\ell}\left(A{B}^{-1}\right)+{\lambda}_{m}\left(H{B}^{-1}\right)}{1+\nu}.$$

- 2.
- Suppose that $A,B,H,E\in {\mathbb{C}}^{n\times n}$ are Hermitian complex matrices of format $n\times n$, that B is positive definite, and that $\nu ={\parallel E\parallel}_{2}/{\lambda}_{n}\left(B\right)<1$. Then we have$$\begin{array}{cc}\hfill {\beta}_{i}\left(A\right){\lambda}_{i}\left(A{B}^{-1}\right)+{\beta}_{n}\left(H\right){\lambda}_{n}\left(H{B}^{-1}\right)\phantom{\rule{1.em}{0ex}}& \le {\lambda}_{i}\left((A+H){(B+H)}^{-1}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le {\alpha}_{i}\left(A\right){\lambda}_{i}\left(A{B}^{-1}\right)+{\alpha}_{1}\left(H\right){\lambda}_{1}\left(H{B}^{-1}\right).\hfill \end{array}$$

#### 2.8. Bounds for Ratio of Bernoulli Numbers

#### 2.9. Special Polynomials

#### 2.10. Complete Monotonicity Properties Related to Polygamma Functions

#### 2.11. Convex Functions and Inequalities

- 1.
- If $-\infty <c<a<b<d<\infty $, the function $f:[c,d]\to \mathbb{R}$ is differentiable, and the derivative $|{f}^{\prime}|$ is convex on $[a,b]$, then we have$$\left|\frac{f\left(a\right)+f\left(b\right)}{2}-\frac{1}{b-a}{\int}_{a}^{b}f\left(x\right)dx\right|\le \frac{b-a}{8}\left(\left|{f}^{\prime}\left(a\right)\right|+\left|{f}^{\prime}\left(b\right)\right|\right).$$
- 2.
- For $0\le a<b<\infty $, if the function $f:[0,b]\to \mathbb{R}$ is m-convex for $m\in (0,1]$ and the Lebesgue integrable, then we have$$\left|\frac{1}{b-a}{\int}_{a}^{b}f\left(x\right)dx\right|\le min\left\{\frac{f\left(a\right)+mf(b/m)}{2},\frac{f\left(b\right)+mf(a/m)}{2}\right\}.$$
- 3.
- For $0\le a<b<\infty $ and $\alpha ,m\in (0,1]$, if the function $f:[0,b]\to \mathbb{R}$ is $(\alpha ,m)$-convex and differentiable and its first derivative is the Lebesgue integrable, then we have$$\begin{array}{c}\left|\frac{f\left(a\right)+f\left(b\right)}{2}-\frac{1}{b-a}{\int}_{a}^{b}f\left(x\right)dx\right|\le \frac{b-a}{2}\frac{1}{{2}^{1-1/q}}\hfill \\ \hfill \times min\left\{{\left[{v}_{1}{\left|{f}^{\prime}\left(a\right)\right|}^{q}+{v}_{2}m{\left|{f}^{\prime}\left(b\right)\right|}^{q}\right]}^{1/q},{\left[{v}_{1}{\left|{f}^{\prime}\left(b\right)\right|}^{q}+{v}_{2}m{\left|{f}^{\prime}\left(a\right)\right|}^{q}\right]}^{1/q}\right\},\end{array}$$$${v}_{1}=\frac{\alpha +1/{2}^{\alpha}}{(\alpha +1)(\alpha +2)}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{v}_{2}=\frac{1}{(\alpha +1)(\alpha +2)}\left(\frac{{\alpha}^{2}+\alpha +2}{2}-\frac{1}{{2}^{\alpha}}\right).$$

#### 2.12. Fractional Derivatives and Integrals

#### 2.13. Differential Geometry

#### 2.14. Pólya Type Integral Inequalities

#### 2.15. Properties of Special Mathematical Means

- 1.
- Let $n\in \mathbb{N}$ be not less than 2 and $\mathbf{a}=({a}_{1},{a}_{2},\cdots ,{a}_{n})$ be a positive sequence, that is, ${a}_{k}>0$ for $1\le k\le n$. The arithmetic and geometric means ${A}_{n}\left(\mathit{a}\right)$ and ${G}_{n}\left(\mathit{a}\right)$ of the positive sequence $\mathit{a}$ are defined, respectively, as$${A}_{n}\left(\mathit{a}\right)=\frac{1}{n}\sum _{k=1}^{n}{a}_{k}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{G}_{n}\left(\mathit{a}\right)={\left(\prod _{k=1}^{n}{a}_{k}\right)}^{1/n}.$$For $z\in \mathbb{C}\setminus (-\infty ,-min\{{a}_{k},1\le k\le n\}]$ and $n\ge 2$, let $\mathit{e}=\left(\stackrel{n}{\overbrace{1,1,\cdots ,1}}\right)$ and$${G}_{n}(\mathit{a}+z\mathit{e})={\left[\prod _{k=1}^{n}({a}_{k}+z)\right]}^{1/n}.$$In (Theorem 1.1 [176]), by virtue of the Cauchy integral formula in the theory of complex functions, the following integral representation was established.Let $\sigma $ be a permutation of the sequence $\{1,2,\cdots ,n\}$ such that the sequence $\sigma \left(\mathit{a}\right)=\left({a}_{\sigma \left(1\right)},{a}_{\sigma \left(2\right)},\cdots ,{a}_{\sigma \left(n\right)}\right)$ is a rearrangement of $\mathit{a}$ in an ascending order ${a}_{\sigma \left(1\right)}\le {a}_{\sigma \left(2\right)}\le \cdots \le {a}_{\sigma \left(n\right)}$. Then the principal branch of the geometric mean ${G}_{n}(\mathit{a}+z\mathit{e})$ has the integral representation$${G}_{n}(\mathit{a}+z\mathit{e})={A}_{n}\left(\mathit{a}\right)+z-\frac{1}{\pi}\sum _{\ell =1}^{n-1}sin\frac{\ell \pi}{n}{\int}_{{a}_{\sigma (\ell )}}^{{a}_{\sigma (\ell +1)}}{\left|\prod _{k=1}^{n}({a}_{k}-t)\right|}^{1/n}\frac{\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t}{t+z}$$for $z\in \mathbb{C}\setminus (-\infty ,-min\{{a}_{k},1\le k\le n\}]$.Taking $z=0$ in the integral representation (12) yields the fundamental inequality$${G}_{n}\left(\mathit{a}\right)={A}_{n}\left(\mathit{a}\right)-\frac{1}{\pi}\sum _{\ell =1}^{n-1}sin\frac{\ell \pi}{n}{\int}_{{a}_{\sigma (\ell )}}^{{a}_{\sigma (\ell +1)}}{\left[\prod _{k=1}^{n}|{a}_{k}-t|\right]}^{1/n}\frac{\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t}{t}\le {A}_{n}\left(\mathit{a}\right).$$For $0<{a}_{1}\le {a}_{2}\le {a}_{3}$, taking $n=2,3$ in (13) gives$$\frac{{a}_{1}+{a}_{2}}{2}-\sqrt{{a}_{1}{a}_{2}}\phantom{\rule{0.166667em}{0ex}}=\frac{1}{\pi}{\int}_{{a}_{1}}^{{a}_{2}}\sqrt{\left(1-\frac{{a}_{1}}{t}\right)\left(\frac{{a}_{2}}{t}-1\right)}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t\ge 0$$$$\frac{{a}_{1}+{a}_{2}+{a}_{3}}{3}-\sqrt[3]{{a}_{1}{a}_{2}{a}_{3}}\phantom{\rule{0.166667em}{0ex}}=\frac{\sqrt{3}\phantom{\rule{0.166667em}{0ex}}}{2\pi}{\int}_{{a}_{1}}^{{a}_{3}}\sqrt[3]{\left|\left(1-\frac{{a}_{1}}{t}\right)\left(1-\frac{{a}_{2}}{t}\right)\left(1-\frac{{a}_{3}}{t}\right)\right|}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t\ge 0.$$These texts are excerpted from the site https://math.stackexchange.com/a/4256320/945479 accessed on 10 July 2022.
- 2.
- The weighted version of the integral representation (12) can be found in the paper (Theorem 3.1 [175]). We recite the weighted version as follows.For $n\ge 2$, $\mathit{a}=({a}_{1},{a}_{2},\cdots ,{a}_{n})$, and $\mathbf{w}=({w}_{1},{w}_{2},\cdots ,{w}_{n})$ with ${a}_{k},{w}_{k}>0$ and ${\sum}_{k=1}^{n}{w}_{k}=1$, the weighted arithmetic and geometric means ${A}_{w,n}\left(\mathit{a}\right)$ and ${G}_{w,n}\left(\mathit{a}\right)$ of $\mathit{a}$ with the positive weight $\mathit{w}$ are defined, respectively, as$${A}_{\mathit{w},n}\left(\mathit{a}\right)=\sum _{k=1}^{n}{w}_{k}{a}_{k}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{G}_{\mathit{w},n}\left(\mathit{a}\right)=\prod _{k=1}^{n}{a}_{k}^{{w}_{k}}.$$Let us denote $\alpha =min\{{a}_{k},1\le k\le n\}$. For a complex variable $z\in \mathbb{C}\setminus (-\infty ,-\alpha ]$, we introduce the complex function$${G}_{\mathit{w},n}(\mathit{a}+z)=\prod _{k=1}^{n}{({a}_{k}+z)}^{{w}_{k}}.$$With the aid of the Cauchy integral formula in the theory of complex functions, the following integral representation was established in (Theorem 3.1 [175]).Let $0<{a}_{k}\le {a}_{k+1}$ for $1\le k\le n-1$ and $z\in \mathbb{C}\setminus (-\infty ,-{a}_{1}]$. Then the principal branch of the weighted geometric mean ${G}_{\mathit{w},n}(\mathit{a}+z)$ with a positive weight $\mathit{w}=({w}_{1},{w}_{2},\cdots ,{w}_{n})$ has the integral representation$$\begin{array}{c}{G}_{\mathit{w},n}(\mathit{a}+z)-{A}_{\mathit{w},n}\left(\mathit{a}\right)\hfill \\ \hfill =z-\frac{1}{\pi}\sum _{\ell =1}^{n-1}sin\left[\left(\sum _{k=1}^{\ell}{w}_{k}\right)\pi \right]{\int}_{{a}_{\ell}}^{{a}_{\ell +1}}\prod _{k=1}^{n}{|{a}_{k}-t|}^{{w}_{k}}\frac{\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t}{t+z}.\end{array}$$Letting $z=0$ in the integral representation (14) gives the fundamental inequality$$\begin{array}{cc}\hfill {G}_{\mathit{w},n}\left(\mathit{a}\right)& ={A}_{\mathit{w},n}\left(\mathit{a}\right)-\frac{1}{\pi}\sum _{\ell =1}^{n-1}sin\left[\left(\sum _{k=1}^{\ell}{w}_{k}\right)\pi \right]{\int}_{{a}_{\ell}}^{{a}_{\ell +1}}\prod _{k=1}^{n}{|{a}_{k}-t|}^{{w}_{k}}\frac{\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t}{t}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le {A}_{\mathit{w},n}\left(\mathit{a}\right).\hfill \end{array}$$Setting $n=2$ in (15) leads to$$\begin{array}{cc}\hfill {a}_{1}^{{w}_{1}}{a}_{2}^{{w}_{2}}& ={w}_{1}{a}_{1}+{w}_{2}{a}_{2}-\frac{sin\left({w}_{1}\pi \right)}{\pi}{\int}_{{a}_{1}}^{{a}_{2}}{\left(1-\frac{{a}_{1}}{t}\right)}^{{w}_{1}}{\left(\frac{{a}_{2}}{t}-1\right)}^{{w}_{2}}\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le {w}_{1}{a}_{1}+{w}_{2}{a}_{2}\hfill \end{array}$$
- 3.
- For ${a}_{k}<{a}_{k+1}$ and ${w}_{k}>0$ with ${\sum}_{k=1}^{n}{w}_{k}=1$ and $n\ge 2$, the principal branch of the reciprocal ${H}_{\mathit{a},\mathit{w},n}\left(z\right)$ of the weighted geometric mean ${G}_{\mathbf{w},n}(\mathit{a}+z)$ can be represented by$$\begin{array}{cc}\hfill {H}_{\mathit{a},\mathit{w},n}\left(z\right)& =\frac{1}{{\prod}_{k=1}^{n}{(z+{a}_{k})}^{{w}_{k}}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{\pi}\sum _{\ell =1}^{n-1}sin\left(\pi \sum _{k=1}^{\ell}{w}_{k}\right){\int}_{{a}_{\ell}}^{{a}_{\ell +1}}\frac{1}{{\prod}_{k=1}^{n}{|t-{a}_{k}|}^{{w}_{k}}}\frac{dt}{t+z},\hfill \end{array}$$

#### 2.16. Invited Visits and Promotions

#### 2.17. Editorial and Refereeing Appointments

- 1.
- Advances in Inequalities and Applications (since 2012);
- 2.
- Advances in Nonlinear Variational Inequalities (since 1998);
- 3.
- Journal of Inequalities and Special Functions (since 2010);
- 4.
- Journal of Inequalities in Pure and Applied Mathematics (since 2000 to 2009);
- 5.
- Journal of Mathematical Inequalities (since 2007);
- 6.
- Mathematical Inequalities and Applications (since 1998);
- 7.
- Turkish Journal of Inequalities (since 2017).

## 3. Statistics of Qi’s Contributions

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Agarwal, R.P.; Karapinar, E.; Kostić, M.; Cao, J.; Du, W.-S.
A Brief Overview and Survey of the Scientific Work by Feng Qi. *Axioms* **2022**, *11*, 385.
https://doi.org/10.3390/axioms11080385

**AMA Style**

Agarwal RP, Karapinar E, Kostić M, Cao J, Du W-S.
A Brief Overview and Survey of the Scientific Work by Feng Qi. *Axioms*. 2022; 11(8):385.
https://doi.org/10.3390/axioms11080385

**Chicago/Turabian Style**

Agarwal, Ravi Prakash, Erdal Karapinar, Marko Kostić, Jian Cao, and Wei-Shih Du.
2022. "A Brief Overview and Survey of the Scientific Work by Feng Qi" *Axioms* 11, no. 8: 385.
https://doi.org/10.3390/axioms11080385