Angiotensin II induces the secretion of ICAM-1 and MCP-1 in human airway smooth muscle cells in vitro

Ning Li; Yating Huo; Haojun Xie; Yuanxiong Cheng; Ning Li; Yating Huo; Haojun Xie; Yuanxiong Cheng

doi:10.3934/medsci.2018.3.259

AIMS Medical Science

2018, Volume 5, Issue 3: 259-267. doi: 10.3934/medsci.2018.3.259

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Research article

Angiotensin II induces the secretion of ICAM-1 and MCP-1 in human airway smooth muscle cells in vitro

1.
Department of Neurosurgery, The First Affiliated Hospital, Jinan University, Guangzhou, Guangdong, China
2.
Department of Respiratory Disease, The Third Affiliated Hospital, Southern Medical University, Guangzhou, China

Received: 23 March 2018 Accepted: 11 June 2018 Published: 27 June 2018

Background: Angiotensin [1] II is known to cause human airway smooth muscle (HASM) contraction and closely relate to asthma, however, the effect of Ang II on HASM cells (HASMCs) secretion function is still unclear. Methods: Primary cultured HASMCs were treated with Ang II, Ang II + Ang-(1–7), and Ang II + irbersatan (IRB), respectively. The nuclear translocation of the NF-κB p65 in HASMCs in each group was analyzed performed with Immunofluorescence. The expression levels of intercellular adhesion molecule-1 (ICAM-1) and monocyte chemoattractant protein-1 (MCP-1) were evaluated by Real-time polymerase chain reaction (PCR) and Enzyme linked immunosorbent assay (ELISA). Results: Ang II caused sharply increase of the nuclear translocation of NF-κB p65, also, Ang II significantly increased expression of ICAM-1 and MCP-1 secreted by HASMCs, this effect could be inhibited by Ang-(1–7) and IRB. Conclusion: These data may indicate the effect of Ang II on inducing HASMCs to secret ICAM-1 and MCP-1, which might be through the NF-κB p65 pathway by AT-1 receptor.

Keywords:

Citation: Ning Li, Yating Huo, Haojun Xie, Yuanxiong Cheng. Angiotensin II induces the secretion of ICAM-1 and MCP-1 in human airway smooth muscle cells in vitro[J]. AIMS Medical Science, 2018, 5(3): 259-267. doi: 10.3934/medsci.2018.3.259

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Abstract

1. Introduction and main results

If $ h $ and $ k $ are integers with $ k > 0 $, the classical Dedekind sums $ S(h, k) $ are defined as

$ S(h,k) = \sum\limits_{a = 1}^{k}\left(\left(\frac{a}{k}\right)\right)\left(\left(\frac{ah}{k}\right) \right), $

where

$ ((x)) = \left\{

$\begin{array}{ll} x-[x]-\frac{1}{2}, &\hbox{if $x$ is not an integer; } \\ 0, &\hbox{if $x$ is an integer. } \end{array}$ \right. $

The various properties of $ S(h, k) $ were investigated by many authors, one of which is reciprocity theorem (see Tom M. Apostol ^[1] or L. Carlitz ^[2]). That is, for all positive integers $ h $ and $ k $ with $ (h, k) = 1 $, we have the identity

$ S(h,k)+S(k,h) = \frac{h^{2}+k^{2}+1}{12hk}-\frac{1}{4}. $

Conrey et al. ^[3] studied the mean value distribution of $ S(h, k) $ and deduced the important asymptotic formula

$\begin{eqnarray*} && \mathop{{\sum}'}^{k}_{h = 1}\left|S(h,k)\right|^{2m} = f_{m}(k)\left( \frac{k}{12} \right)^{2m}+O\left(\left(k^{\frac{9}{5}}+k^{2m-1+\frac{1}{m+1}} \right)\log^{3}k \right), \end{eqnarray*}$ $

where $

$\begin{eqnarray*} \mathop{{\sum}'}^{k}_{h = 1}\end{eqnarray*}$ $ denotes the summation over all $ h $ such that $ (h, k) = 1 $ and

$\begin{eqnarray*} && \sum\limits_{n = 1}^{\infty}\frac{f_{m}(n)}{n^{s}} = 2\frac{\zeta^{2}(2m)}{\zeta(4m)}\cdot\frac{\zeta(s+4m-1)}{\zeta^{2}(s+2m)}\zeta(s). \end{eqnarray*}$ $

Moreover, X. L. He and W. P. Zhang ^[4] gave an interesting asymptotic formula for the Dedekind sums with a weight of Hurwitz zeta-function as follows:

$\begin{eqnarray*} && \mathop{{\sum}'}^{k}_{h = 1}\zeta^{2}\left(\frac{1}{2},\frac{h}{k}\right)S^{2}(h,k) = \frac{k^{3}}{144}\zeta(3)\prod\limits_{p|k}\left( 1-\frac{1}{p^{3}}\right)+O\left( k^{\frac{5}{2}}\exp\left(\frac{3\log k}{\log\log k} \right) \right). \end{eqnarray*}$ $

Other sums analogous to the Dedekind sums are the Hardy sums. Using the notation of Berndt and Goldberg ^[5], they defined

$ S_{1}(h,k) = \sum\limits_{j = 1}^{k-1}(-1)^{j+1+\left[\frac{hj}{k}\right]}, $

where $ h $ and $ k $ are integers with $ k > 0 $.

In 2014, H. Zhang and W. P. Zhang ^[6] obtained some beautiful identities involving $ S_{1}(h, k) $ in the forms of

$ \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}K(m,p)K(n,p)S_{1}(2m\overline{n},p), $

$ \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,p)\right|^{2}\left|K(n,p)\right|^{2}S_{1}(2m\overline{n},p), $

where $ K(n, p) $ denotes the reduced form of the general Kloosterman sums attached to a Dirichlet character $ \lambda $ modulo $ k $ as

$\begin{eqnarray*} && K(r,l,\lambda;k) = \mathop{{\sum}'}^{k}_{a = 1}\lambda(a)e\left(\frac{ra+l\overline{a}}{k}\right), \end{eqnarray*}$ $

where $ e(x) = e^{2 \pi i x} $, $ \overline{a} $ denotes the solution of the congruence $ x \cdot a\equiv 1 \bmod k $.

Recently, H. F. Zhang and T. P. Zhang ^[7] extended the results in ^[6] to a more general situation as

$ \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}K(m,s,\lambda;p)\overline{K(n,t,\lambda;p)}S_{1}(2m\overline{n},p), $

$ \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{1}(2m\overline{n},p), $

where $ \overline{K(n, t, \lambda; p)} $ denotes complex conjugate of $ K(n, t, \lambda; p) $.

Actually there are six forms of Hardy sums (see Berndt ^[8] and Goldberg ^[9]). A natural question is whether we can obtain similar results by replacing $ S_{1}(h, k) $ with other forms of Hardy sums. Due to some technical reasons, for most of other forms of Hardy sums, the answer is no! Thanks to the important relationships among Hardy sums and Dedekind sums built by R. Sitaramachandrarao ^[10], we are lucky to find the only one $ S_{3}(h, p) $ to replace, with

$\begin{eqnarray*} S_{3}(h,k) = \sum\limits_{j = 1}^{k}(-1)^{j}\left(\left(\frac{hj}{k}\right)\right). \end{eqnarray*}$ $

Our starting point relies heavily on the following in ^[10] as:

Proposition 1. Let $ k $ be an odd positive integer, $ h $ be an integer with $ (h, k) = 1 $. Then

$\begin{eqnarray*} && S_{3}(h,k) = 2S(h,k)-4S(2h,k). \end{eqnarray*}$ $

Then applying the properties of Gauss sums and the mean square value of Dirichlet $ L $-functions, we have

Theorem 1. Let $ p $ be an odd prime. Then for any Dirichlet character $ \lambda \bmod p $ and any integer $ s $, $ t $ with $ (s, p) = (t, p) = 1 $, we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}K(m,s,\lambda;p)\overline{K(n,t,\lambda;p)}S_{3}(m\overline{n},p) = \left\{ \begin{array}{ll} \frac{p-1}{2}, & \hbox{if $\overline{\lambda}\chi = \chi_{0}$}; \\ \frac{p(p-1)}{2}, & \hbox{if $\overline{\lambda}\chi\neq\chi_{0}$}, \end{array} \right. \end{eqnarray*}$ $

where $ \chi $ is an odd Dirichlet character modulo $ p $ and $ \chi_{0} $ is the principal character modulo $ p $.

Theorem 2. Let $ p $ be an odd prime with $ p\equiv 1 \bmod 4 $. Then for any Dirichlet character $ \lambda \bmod p $ and any integer $ s $, $ t $ with $ (s, p) = (t, p) = 1 $, we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{3}(m\overline{n},p)\\ &&\left\{ \begin{array}{ll} = \frac{p^{2}(p-1)}{2}, & \hbox{if $\overline{\lambda\chi}\neq\chi_{0},\ \overline{\lambda}\chi\neq\chi_{0}$};\\ = \frac{p(p-1)}{2}, & \hbox{if $\overline{\lambda\chi}\neq\chi_{0},\ \overline{\lambda}\chi = \chi_{0}$};\\ \leq p^{\frac{9}{2}}+\frac{1}{2}p^{4}-4p^{\frac{7}{2}}-p^{3}+5p^{\frac{5}{2}}+p^{2}-2p^{\frac{3}{2}}-\frac{1}{2}p, & \hbox{if $\overline{\lambda\chi} = \chi_{0},\ \overline{\lambda}\chi = \chi_{0}$}; \\ \leq p^{5}-3p^{4}+3p^{3}-\frac{1}{2}p^{2}-\frac{1}{2}p, & \hbox{if $\overline{\lambda\chi} = \chi_{0},\ \overline{\lambda}\chi\neq\chi_{0}$}. \end{array} \right. \end{eqnarray*}$ $

Theorem 3. Let $ p $ be an odd prime with $ p\equiv 3 \bmod 8 $. Then for any Dirichlet character $ \lambda \bmod p $ and any integer $ s $, $ t $ with $ (s, p) = (t, p) = 1 $, we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{3}(m\overline{n},p)\\ &&\left\{ \begin{array}{ll} = \frac{p^{2}(p-1)}{2}-6p^{2}h_{p}^{2}, & \hbox{if $\overline{\lambda\chi}\neq\chi_{0},\ \overline{\lambda}\chi\neq\chi_{0}$};\\ = \frac{p(p-1)}{2}-6ph_{p}^{2}, & \hbox{if $\overline{\lambda\chi}\neq\chi_{0},\ \overline{\lambda}\chi = \chi_{0}$};\\ \leq p^{\frac{9}{2}}+\frac{1}{2}p^{4}-4p^{\frac{7}{2}}-p^{3}+5p^{\frac{5}{2}}+p^{2}-2p^{\frac{3}{2}}-\frac{1}{2}p+6p^{2}h_{p}^{2}, & \hbox{if $\overline{\lambda\chi} = \chi_{0},\ \overline{\lambda}\chi = \chi_{0}$}; \\ \leq p^{5}-3p^{4}+3p^{3}-\frac{1}{2}p^{2}-\frac{1}{2}p+6p^{3}h_{p}^{2}, & \hbox{if $\overline{\lambda\chi} = \chi_{0},\ \overline{\lambda}\chi\neq\chi_{0}$}, \end{array} \right. \end{eqnarray*}$ $

where $ h_{p} $ denotes the class number of the quadratic field $ Q\left(\sqrt{-p}\right) $.

Theorem 4. Let $ p $ be an odd prime with $ p\equiv 7 \bmod 8 $. Then for any Dirichlet character $ \lambda \bmod p $ and any integer $ s $, $ t $ with $ (s, p) = (t, p) = 1 $, we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{3}(m\overline{n},p)\\ &&\left\{ \begin{array}{ll} = \frac{p^{2}(p-1)}{2}+2p^{2}h_{p}^{2}, & \hbox{if $\overline{\lambda\chi}\neq\chi_{0},\ \overline{\lambda}\chi\neq\chi_{0}$};\\ = \frac{p(p-1)}{2}+2ph_{p}^{2}, & \hbox{if $\overline{\lambda\chi}\neq\chi_{0},\ \overline{\lambda}\chi = \chi_{0}$};\\ \leq p^{\frac{9}{2}}+\frac{1}{2}p^{4}-4p^{\frac{7}{2}}-p^{3}+5p^{\frac{5}{2}}+p^{2}-2p^{\frac{3}{2}}-\frac{1}{2}p+2p^{2}h_{p}^{2}, & \hbox{if $\overline{\lambda\chi} = \chi_{0},\ \overline{\lambda}\chi = \chi_{0}$}; \\ \leq p^{5}-3p^{4}+3p^{3}-\frac{1}{2}p^{2}-\frac{1}{2}p+2p^{3}h_{p}^{2}, & \hbox{if $\overline{\lambda\chi} = \chi_{0},\ \overline{\lambda}\chi\neq\chi_{0}$}. \end{array} \right. \end{eqnarray*}$ $

Taking $ \lambda = \lambda_{0} $, $ s = t = 1 $ in Theorems 1–4, we immediately deduce the following results.

Corollary 1. Let $ p $ be an odd prime. Then we have the identity

$ \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}K(m,p)K(n,p)S_{3}(m\overline{n},p) = \frac{p(p-1)}{2}. $

Corollary 2. Let $ p $ be an odd prime. Then we have

$ \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,p)\right|^{2}\left|K(n,p)\right|^{2}S_{3}(m\overline{n},p) = \left\{

$\begin{array}{ll} \frac{p^{2}(p-1)}{2}, & \hbox{if $p\equiv 1 \bmod 4$}; \\ \frac{p^{2}(p-1)}{2}-6p^{2}h_{p}^{2}, & \hbox{if $p\equiv 3 \bmod 8$};\\ \frac{p^{2}(p-1)}{2}+2p^{2}h_{p}^{2}, & \hbox{if $p\equiv 7 \bmod 8$}.\\ \end{array}$ \right. $

2. Some Lemmas

To prove the Theorems, we need the following Lemmas.

Lemma 1. Let $ k > 2 $ be an integer. Then for any integer $ a $ with $ (a, k) = 1, $ we have the identity

$ S(a,k) = \frac{1}{\pi^2k}\sum\limits_{d\mid k}\frac{d^2}{\phi(d)}\mathop{\sum\limits_{\chi \bmod d}}_{\chi(-1) = -1}\chi(a)\left|L(1,\chi)\right|^2, $

where $ L(1, \chi) $ denotes the Dirichlet $ L $-function corresponding to Dirichlet character $ \chi $ mod $ d $.

Proof. See Lemma 2 of ^[11].

Lemma 2. Let $ p $ be an odd prime, $ s $ be any integer with $ (s, p) = 1 $. Then for any non-principal character $ \chi \bmod p $ and any Dirichlet character $ \lambda \bmod p $, we have

$ \left|\sum\limits_{m = 1}^{p-1}\chi(m)K(m,s,\lambda;p)\right| = \left\{

$\begin{array}{ll} p^{{\frac{1}{2}}}, & \hbox{if $\overline{\lambda}\chi = \chi_{0}$}; \\ p, & \hbox{if $\overline{\lambda}\chi\neq\chi_{0}$}. \end{array}$ \right. $

Proof. See Lemma 2 of reference ^[7].

Lemma 3. Let $ p $ be an odd prime, $ s $ be any integer with $ (s, p) = 1 $. Then for any non-principal character $ \chi \bmod p $ and any Dirichlet character $ \lambda \bmod p $, we have

$\begin{eqnarray*} && \left|\sum\limits_{m = 1}^{p-1} \chi(m)\left|K(m,s,\lambda;p)\right|^{2}\right| = \left\{ \begin{array}{ll} p\left|\tau\left(\overline{\chi}^{2}\right)\right|, & \hbox{if $\overline{\lambda\chi}\neq\chi_{0},\ \overline{\lambda}\chi\neq\chi_{0}$};\\ p^{\frac{1}{2}}\left|\tau\left(\overline{\chi}^{2}\right)\right|, & \hbox{if $\overline{\lambda\chi}\neq\chi_{0},\ \overline{\lambda}\chi = \chi_{0}$}; \\ p\left|\tau\left(\overline{\chi}^{2}\right)+(p-1)\right|, & \hbox{if $\overline{\lambda\chi} = \chi_{0},\ \overline{\lambda}\chi = \chi_{0}$}; \\ p\left|-\tau\left(\overline{\chi}^{2}\right)\tau\left(\overline{\lambda}\chi\right)+(p-1)\right|, & \hbox{if $\overline{\lambda\chi} = \chi_{0},\ \overline{\lambda}\chi\neq\chi_{0}$}, \end{array} \right. \end{eqnarray*}$ $

where $ \tau(\chi) = \sum_{a = 1}^{p}\chi(a) e\left(\frac{a}{p}\right) $ denotes the classical Gauss sums.

Proof. See Lemma 1 of reference ^[7].

Lemma 4. Let $ p $ be an odd prime, then we have

$\begin{eqnarray*} && \mathop{\sum\limits_{\chi \bmod p}}_{\chi(-1) = -1}\left|L(1,\chi)\right|^2 = \frac{\pi^2}{12}\cdot\frac{(p-1)^2(p-2)}{p^2} , \end{eqnarray*}$ $

$\begin{eqnarray*} && \mathop{\sum\limits_{\chi \bmod p}}_{\chi(-1) = -1}\chi(2)\cdot\left|L(1,\chi)\right|^2 = \frac{\pi^2}{24}\cdot\frac{(p-1)^2(p-5)}{p^2}. \end{eqnarray*}$ $

Proof. See Lemma 5 of reference ^[6].

3. Proof of Theorems

Now we come to prove our Theorems.

Firstly, we prove Theorem 1. Applying Proposition $ 1 $ and Lemma $ 1 $, we obtain

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}K(m,s,\lambda;p)\overline{K(n,t,\lambda;p)}S_{3}(m\overline{n},p)\\ && = \frac{2p}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\sum\limits_{m = 1}^{p-1}\chi(m) K(m,s,\lambda;p)\cdot\sum\limits_{n = 1}^{p-1}\chi(\overline{n})\overline{K(n,t,\lambda;p)}\cdot|L(1,\chi)|^2\\ &&\quad-\frac{4p}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2)\sum\limits_{m = 1}^{p-1}\chi(m) K(m,s,\lambda;p)\cdot\sum\limits_{n = 1}^{p-1}\chi(\overline{n})\overline{K(n,t,\lambda;p)}\cdot|L(1,\chi)|^2\\ && = \frac{2p}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\left|\sum\limits_{m = 1}^{p-1}\chi(m)K(m,s,\lambda;p)\right|^{2}\left|L(1,\chi)\right|^2\\ &&\quad-\frac{4p}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1} \chi(2)\left|\sum\limits_{m = 1}^{p-1}\chi(m)K(m,s,\lambda;p)\right|^{2}\left|L(1,\chi)\right|^2. \end{eqnarray*}$ $

Then from Lemma $ 2 $ and Lemma $ 4 $, if $ \overline{\lambda}\chi = \chi_{0} $, we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}K(m,s,\lambda;p)\overline{K(n,t,\lambda;p)}S_{3}(m\overline{n},p)\\ && = \frac{2p^{2}}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\left|L(1,\chi)\right|^2 -\frac{4p^{2}}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2)\left|L(1,\chi)\right|^2\\ && = \frac{2p^{2}}{\pi^2(p-1)}\cdot\frac{\pi^{2}}{12}\frac{(p-1)^{2}(p-2)}{p^{2}} -\frac{4p^{2}}{\pi^2(p-1)}\cdot\frac{\pi^{2}}{24}\frac{(p-1)^{2}(p-5)}{p^{2}}\\ && = \frac{p-1}{2}. \end{eqnarray*}$ $

While if $ \overline{\lambda}\chi\neq\chi_{0} $, we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}K(m,s,\lambda;p)\overline{K(n,t,\lambda;p)}S_{3}(m\overline{n},p)\\ && = \frac{2p^{3}}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\left|L(1,\chi)\right|^2 -\frac{4p^{3}}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2)\left|L(1,\chi)\right|^2\\ && = \frac{2p^{3}}{\pi^2(p-1)}\cdot\frac{\pi^{2}}{12}\cdot\frac{(p-1)^{2}(p-2)}{p^{2}} -\frac{4p^{3}}{\pi^2(p-1)}\cdot\frac{\pi^{2}}{24}\cdot\frac{(p-1)^{2}(p-5)}{p^{2}}\\ && = \frac{p(p-1)}{2}. \end{eqnarray*}$ $

This completes the proof of Theorem 1.

Then we prove Theorem 2. From Proposition $ 1 $ and Lemma $ 1 $, we obtain

Since $ p\equiv 1 \bmod 4 $, and notice that $ \left|\tau\left(\overline{\chi}^{2}\right)\right| = \sqrt{p} $. From Lemma $ 3 $ and Lemma $ 4 $, if $ \overline{\lambda\chi}\neq\chi_{0} $, $ \overline{\lambda}\chi\neq\chi_{0} $, we have

Similarly, if $ \overline{\lambda\chi}\neq\chi_{0}, \ \overline{\lambda}\chi = \chi_{0} $, we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{3}(m\overline{n},p) = \frac{p(p-1)}{2}. \end{eqnarray*}$ $

If $ \overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi = \chi_{0} $, we can obtain

$\begin{eqnarray*} && \left|\sum\limits_{m = 1}^{p-1}\chi(m)\left|K(m,s,\lambda;p)\right|^{2}\right|^{2}\\ && = p^{2}\left[ \left(Re\ \tau\left(\overline{\chi}^{2}\right)+(p-1)\right)^{2}+\left(Im\ \tau\left(\overline{\chi}^{2}\right)\right)^{2}\right]\\ && = p^{2}\left[p+2(p-1)Re\ \tau\left(\overline{\chi}^{2}\right)+(p-1)^{2}\right]. \end{eqnarray*}$ $

So we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{3}(m\overline{n},p)\\ && = \frac{2p^{3}}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\left[p+2(p-1)Re\ \tau\left(\overline{\chi}^{2}\right)+(p-1)^{2}\right]\left|L(1,\chi)\right|^2\\ &&\quad-\frac{4p^{3}}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2)\left[p+2(p-1)Re\ \tau\left(\overline{\chi}^{2}\right)+(p-1)^{2}\right]\left|L(1,\chi)\right|^2\\ && = \frac{2p^{3}[p+(p-1)^{2}]}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\left|L(1,\chi)\right|^2+\frac{4p^{3}}{\pi^2}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}Re\ \left(\tau\left(\overline{\chi}^{2}\right)\right)\left|L(1,\chi)\right|^2\\ &&\quad -\frac{4p^{3}[p+(p-1)^{2}]}{\pi^2(p-1)} \sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2)\left|L(1,\chi)\right|^2-\frac{8p^{3}}{\pi^2}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2)Re\ \left(\tau\left(\overline{\chi}^{2}\right)\right)\left|L(1,\chi)\right|^2\\ && = \frac{p(p-1)(p^{2}-p+1)}{2}+\frac{4p^{3}}{\pi^{2}}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1} Re\ \left(\tau \left(\overline{\chi}^{2}\right)\right) \left|L(1,\chi)\right|^{2}\\ &&\quad -\frac{8p^{3}}{\pi^{2}}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2) Re\ \left(\tau\left(\overline{\chi}^{2}\right)\right)\left|L(1,\chi)\right|^{2}. \end{eqnarray*}$ $

Noting that $ x\leq |x| $ holds for any real number $ x $, we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{3}(m\overline{n},p)\\ &&\leq\left|\sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{3}(m\overline{n},p)\right|\\ &&\leq\frac{p(p-1)(p^{2}-p+1)}{2}+\frac{4p^{\frac{7}{2}}}{\pi^{2}}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\left|L(1,\chi)\right|^{2}+\frac{8p^{\frac{7}{2}}}{\pi^{2}}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\left|L(1,\chi)\right|^{2}\\ && = \frac{p(p-1)(p^{2}-p+1)}{2}+\frac{4p^{\frac{7}{2}}}{\pi^{2}}\cdot\frac{\pi^{2}}{12}\cdot\frac{(p-1)^{2}(p-2)}{p^{2}}+ \frac{8p^{\frac{7}{2}}}{\pi^{2}}\cdot\frac{\pi^{2}}{12}\cdot\frac{(p-1)^{2}(p-2)}{p^{2}}\\ && = p^{\frac{9}{2}}+\frac{1}{2}p^{4}-4p^{\frac{7}{2}}-p^{3}+5p^{\frac{5}{2}}+p^{2}-2p^{\frac{3}{2}}-\frac{1}{2}p. \end{eqnarray*}$ $

Similarly, if $ \overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi\neq\chi_{0} $, we have

This completes the proof of Theorem 2.

Next we turn to prove Theorem 3. Since $ p\equiv 3 \bmod 4 $, note that $ \left(\frac{-1}{p}\right) = \chi_{2}(-1) = -1 $, $ L(1, \chi_{2}) = \frac{\pi\cdot h_{p}}{\sqrt{p}} $, and $ \tau\left(\overline{\chi}^{2}_{2}\right) = -1 $. From Lemma $ 3 $ and Lemma $ 4 $, if $ \overline{\lambda\chi}\neq\chi_{0}, \ \overline{\lambda}\chi\neq\chi_{0} $, we have

Similarly, if $ \overline{\lambda\chi}\neq\chi_{0}, \ \overline{\lambda}\chi = \chi_{0} $, we have

If $ \overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi = \chi_{0} $, we can obtain

$\begin{eqnarray*} && p^{2}\left|\tau\left(\overline{\chi}^{2}_{2}\right)+(p-1)\right|^{2} = p^{2}\left[1+2(p-1)Re\ \tau\left(\overline{\chi}^{2}_{2}\right)+(p-1)^{2}\right]. \end{eqnarray*}$ $

So we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{3}(m\overline{n},p)\\ && = \frac{2p^{3}}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\left[p+2(p-1)Re\ \tau\left(\overline{\chi}^{2}\right)+(p-1)^{2}\right]\left|L(1,\chi)\right|^2\\ &&\quad-\frac{2p^{3}}{\pi^2(p-1)}\left[p+2(p-1)Re\ \tau\left(\overline{\chi}^{2}_{2}\right)+(p-1)^{2}\right]\left|L(1,\chi_{2})\right|^2\\ &&\quad+\frac{2p^{3}}{\pi^2(p-1)}\left[1+2(p-1)Re\ \tau\left(\overline{\chi}^{2}_{2}\right)+(p-1)^{2}\right]\left|L(1,\chi_{2})\right|^2\\ &&\quad-\frac{4p^{3}}{\pi^2(p-1)}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2)\left[p+2(p-1)Re\ \tau\left(\overline{\chi}^{2}\right)+(p-1)^{2}\right]\left|L(1,\chi)\right|^2\\ &&\quad+\frac{4p^{3}}{\pi^2(p-1)}\left[p+2(p-1)Re\ \tau\left(\overline{\chi}^{2}_{2}\right)+(p-1)^{2}\right]\chi_{2}(2)\left|L(1,\chi_{2})\right|^2\\ &&\quad-\frac{4p^{3}}{\pi^2(p-1)}\left[1+2(p-1)Re\ \tau\left(\overline{\chi}^{2}_{2}\right)+(p-1)^{2}\right]\chi_{2}(2)\left|L(1,\chi_{2})\right|^2\\ && = \frac{p(p-1)(p^{2}-p+1)}{2}+\frac{4p^{3}}{\pi^{2}}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}Re\ \left(\tau\left(\overline{\chi}^{2}\right)\right)\left|L(1,\chi)\right|^{2}\\ &&\quad-\frac{8p^{3}}{\pi^{2}}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2) Re\ \left(\tau\left(\overline{\chi}^{2}\right)\right)\left|L(1,\chi)\right|^{2}-2p^{2}h_{p}^{2}+4p^{2}h_{p}^{2}\left(\frac{2}{p}\right). \end{eqnarray*}$ $

Similarly, if $ \overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi\neq\chi_{0} $, we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}|K(m,s,\lambda;p)|^{2}|K(n,t,\lambda;p)|^{2}S_{3}(m\overline{n},p)\\ && = \frac{p(p-1)(2p^{2}-2p+1)}{2}-\frac{4p^{3}}{\pi^{2}}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}Re\ \left(\tau(\overline{\chi}^{2})\tau(\overline{\lambda}\chi)\right)|L(1,\chi)|^{2} \\ &&\quad+\frac{8p^{3}}{\pi^{2}}\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2) Re\ \left(\tau(\overline{\chi}^{2})\tau(\overline{\lambda}\chi)\right)|L(1,\chi)|^{2}-2p^{3}h_{p}^{2}+4p^{3}h_{p}^{2}\left(\frac{2}{p}\right). \end{eqnarray*}$ $

Combining the fact that

$\begin{eqnarray*} && \left(\frac{2}{p}\right) = (-1)^{\frac{p^{2}-1}{8}} = \left\{ \begin{array}{ll} 1, & \hbox{if $p\equiv\pm 1 \bmod 8$}; \\ -1, & \hbox{if $p\equiv\pm 3 \bmod 8$}, \end{array} \right. \end{eqnarray*}$ $

we deduce that if $ p\equiv 3 \bmod 8 $, then

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{3}(m\overline{n},p)\\ && = \left\{ \begin{array}{ll} \frac{p^{2}(p-1)}{2}-6p^{2}h_{p}^{2}, & \hbox{if $\overline{\lambda\chi}\neq\chi_{0},\ \overline{\lambda}\chi\neq\chi_{0}$};\\ \frac{p(p-1)}{2}-6ph_{p}^{2}, & \hbox{if $\overline{\lambda\chi}\neq\chi_{0},\ \overline{\lambda}\chi = \chi_{0}$}. \end{array} \right. \end{eqnarray*}$ $

If $ \overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi = \chi_{0} $, we have

Then

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}\left|K(m,s,\lambda;p)\right|^{2}\left|K(n,t,\lambda;p)\right|^{2}S_{3}(m\overline{n},p)\\ &&\leq \frac{p(p-1)(p^{2}-p+1)}{2}+\frac{4p^{3}}{\pi^{2}}\left|\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}Re\ \left(\tau\left(\overline{\chi}^{2}\right)\right)\left|L(1,\chi)\right|^{2} \right|\\ &&\quad+\frac{8p^{3}}{\pi^{2}}\left|\sum\limits_{\chi \bmod p \atop \chi(-1) = -1}\chi(2) Re\ \left(\tau\left(\overline{\chi}^{2}\right)\right)\left|L(1,\chi)\right|^{2}\right|+6p^{2}h_{p}^{2}\\ && = p^{\frac{9}{2}}+\frac{1}{2}p^{4}-4p^{\frac{7}{2}}-p^{3}+5p^{\frac{5}{2}}+p^{2}-2p^{\frac{3}{2}}-\frac{1}{2}p+6p^{2}h_{p}^{2}. \end{eqnarray*}$ $

Similarly, if $ \overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi\neq\chi_{0} $, we have

$\begin{eqnarray*} && \sum\limits_{m = 1}^{p-1}\sum\limits_{n = 1}^{p-1}|K(m,s,\lambda;p)|^{2}|K(n,t,\lambda;p)|^{2}S_{3}(m\overline{n},p) \leq p^{5}-3p^{4}+3p^{3}-\frac{1}{2}p^{2}-\frac{1}{2}p+6p^{3}h_{p}^{2}. \end{eqnarray*}$ $

This completes the proof of Theorem 3.

Theorem 4 can be derived by the same method. This completes the proof of our Theorems.

4. Conclusions

In this paper, we obtain some exact computational formulas or upper bounds for hybrid mean value involving Hardy sums and Kloosterman sums (both classical Kloosterman sums and general Kloosterman sums) by applying the properties of Gauss sums and the mean value of Dirichlet $ L $-function. But in some cases, unluckily, it is difficult to get the exact formula. So how to get the exact formula in all cases remains open.

Acknowledgments

The authors want to show their great thanks to the anonymous referee for his/her helpful comments and suggestions.

This work is supported by the National Natural Science Foundation of China (No. 11871317, 11926325, 11926321) and the Fundamental Research Funds for the Central Universities (No. GK201802011).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

[1]	Kaiya H, Kangawa K, Miyazato M (2013) What is the general action of ghrelin for vertebrates? -comparisons of ghrelin's effects across vertebrates. Gen Comp Endocrinol 181: 187-191.
[2]	Braman SS (2006) The global burden of asthma. Chest 130: 4s-12s. doi: 10.1378/chest.130.1_suppl.4S
[3]	Croisant S (2014) Epidemiology of asthma: prevalence and burden of disease. Adv Exp Med Biol 795: 17-29. doi: 10.1007/978-1-4614-8603-9_2
[4]	Holgate ST (2011) Asthma: a simple concept but in reality a complex disease. Eur J Clin Invest 41: 1339-1352. doi: 10.1111/j.1365-2362.2011.02534.x
[5]	Webley WC, Hahn DL (2017) Infection-mediated asthma: etiology, mechanisms and treatment options, with focus on Chlamydia pneumoniae and macrolides. Respir Res 18: 98. doi: 10.1186/s12931-017-0584-z
[6]	Schuijs MJ, Willart MA, Hammad H, et al. (2013) Cytokine targets in airway inflammation. Curr Opin Pharmacol 13: 351-361. doi: 10.1016/j.coph.2013.03.013
[7]	Tliba O, Amrani Y, Panettieri RA, et al. (2008) Is airway smooth muscle the "missing link" modulating airway inflammation in asthma? Chest 133: 236-242. doi: 10.1378/chest.07-0262
[8]	Singh SR, Sutcliffe A, Kaur D, et al. (2014) CCL2 release by airway smooth muscle is increased in asthma and promotes fibrocyte migration. Allergy 69: 1189-1197. doi: 10.1111/all.12444
[9]	McKay S, Sharma HS (2002) Autocrine regulation of asthmatic airway inflammation: role of airway smooth muscle. Respir Res 3: 11.
[10]	Koziol-White CJ, Panettieri RA, et al. (2011) Airway smooth muscle and immunomodulation in acute exacerbations of airway disease. Immunol Rev 242: 178-185. doi: 10.1111/j.1600-065X.2011.01022.x
[11]	Ozier A, Allard B, Bara I, et al. (2011) The pivotal role of airway smooth muscle in asthma pathophysiology. J Allergy (Cairo) 2011: 742710.
[12]	Zheng Y, Tang L, Huang W, et al. (2015) Anti-Inflammatory Effects of Ang-(1–7) in Ameliorating HFD-Induced Renal Injury through LDLr-SREBP2-SCAP Pathway. PLoS One 10: e0136187. doi: 10.1371/journal.pone.0136187
[13]	Padda RS, Shi Y, Lo CS, et al. (2015) Angiotensin-(1–7): A Novel Peptide to Treat Hypertension and Nephropathy in Diabetes? J Diabetes Metab 6.
[14]	Ruiz-Ortega M, Esteban V, Ruperez M, et al. (2006) Renal and vascular hypertension-induced inflammation: role of angiotensin II. Curr Opin Nephrol Hypertens 15: 159-166. doi: 10.1097/01.mnh.0000203190.34643.d4
[15]	Li C, He J, Zhong X, et al. (2018) CX3CL1/CX3CR1 Axis Contributes to Angiotensin II-Induced Vascular Smooth Muscle Cell Proliferation and Inflammatory Cytokine Production. Inflammation.
[16]	Li HY, Yang M, Li Z, et al. (2017) Curcumin inhibits angiotensin II-induced inflammation and proliferation of rat vascular smooth muscle cells by elevating PPAR-gamma activity and reducing oxidative stress. Int J Mol Med 39: 1307-1316. doi: 10.3892/ijmm.2017.2924
[17]	Bihl JC, Zhang C, Zhao Y, et al. (2015) Angiotensin-(1–7) counteracts the effects of Ang II on vascular smooth muscle cells, vascular remodeling and hemorrhagic stroke: Role of the NFsmall ka, CyrillicB inflammatory pathway. Vascul Pharmacol 73: 115-123. doi: 10.1016/j.vph.2015.08.007
[18]	Pang L, Nie M, Corbett L, et al. (2006) Mast cell beta-tryptase selectively cleaves eotaxin and RANTES and abrogates their eosinophil chemotactic activities. J Immunol 176: 3788-3795. doi: 10.4049/jimmunol.176.6.3788
[19]	Sutcliffe AM, Clarke DL, Bradbury DA, et al. (2009) Transcriptional regulation of monocyte chemotactic protein-1 release by endothelin-1 in human airway smooth muscle cells involves NF-kappaB and AP-1. Br J Pharmacol 157: 436-450. doi: 10.1111/j.1476-5381.2009.00143.x
[20]	Deacon K, Knox AJ (2015) Human airway smooth muscle cells secrete amphiregulin via bradykinin/COX-2/PGE2, inducing COX-2, CXCL8, and VEGF expression in airway epithelial cells. Am J Physiol Lung Cell Mol Physiol 309: L237-249. doi: 10.1152/ajplung.00390.2014
[21]	Gangur V, Oppenheim JJ (2000) Are chemokines essential or secondary participants in allergic responses? Ann Allergy Asthma Immunol 84: 569-579. doi: 10.1016/S1081-1206(10)62403-9
[22]	Sousa AR, Lane SJ, Nakhosteen JA, et al. (1994) Increased expression of the monocyte chemoattractant protein-1 in bronchial tissue from asthmatic subjects. Am J Respir Cell Mol Biol 10: 142-147. doi: 10.1165/ajrcmb.10.2.8110469
[23]	Watson ML, Grix SP, Jordan NJ, et al. (1998) Interleukin 8 and monocyte chemoattractant protein 1 production by cultured human airway smooth muscle cells. Cytokine 10: 346-352. doi: 10.1006/cyto.1997.0350
[24]	Nie M, Corbett L, Knox AJ, et al. (2005) Differential regulation of chemokine expression by peroxisome proliferator-activated receptor gamma agonists: interactions with glucocorticoids and beta2-agonists. J Biol Chem 280: 2550-2561. doi: 10.1074/jbc.M410616200
[25]	Wang YX, Ji ML, Jiang CY, et al. (2016) Upregulation of ICAM-1 and IL-1beta protein expression promotes lung injury in chronic obstructive pulmonary disease. Genet Mol Res 15.
[26]	Liang B, Wang X, Zhang N, et al. (2015) Angiotensin-(1–7) Attenuates Angiotensin II-Induced ICAM-1, VCAM-1, and MCP-1 Expression via the MAS Receptor Through Suppression of P38 and NF-kappaB Pathways in HUVECs. Cell Physiol Biochem 35: 2472-2482. doi: 10.1159/000374047

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