摘要：
This paper is concerned with mean square exponential synchronization for two classes of Markovian switching complex networks (MSCNs) under linear negative control scheme from synchronization control cost viewpoint. Two classes of MSCNs are considered. In order to analyze synchronization controllability for the two classes of MSCNs from synchronization control cost viewpoint, synchronization control cost and synchronization controllability are defined, respectively. By utilizing M-matrix properties, a novel Lyapunov function and the stochastic analysis techniques, sufficient criteria of mean square synchronization and global synchronization for the two classes of MSCNs under feedback control are derived. Three numerical examples and simulations are given to illustrate the following conclusions: (1) the two classes of MSCNs under the same linear negative feedback controller can achieve mean square exponential synchronization and global synchronization if they satisfy sufficient conditions of mean square exponential synchronization; (2) by computing synchronization control cost, we obtain that global synchronization controllability for the first class of MSCNs may perhaps stronger or poorer than that of the second class of MSCNs with time-varying delays. (C) 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
期刊：
JOURNAL OF INEQUALITIES AND APPLICATIONS,2015年2015(1):1-14 ISSN：1029-242X
通讯作者：
Chu, Yu-Ming
作者机构：
[Chu, Yu-Ming; Yang, Zhen-Hang] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
通讯机构：
[Chu, Yu-Ming] H;Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
关键词：
Mills ratio;monotonicity;convexity;bound
摘要：
In the article, we present several sharp bounds for the Mills ratio
$R(x)=e^{x^{2}/2}\int_{x}^{\infty}e^{-t^{2}/2}\,dt$
(
$x>0$
) in terms of the functions
$I_{a}(x)=a/[\sqrt{x^{2}+2a}+(a-1)x]$
and
$J(x)=a/[\sqrt{x^{2}+2a^{2}/\pi}+2ax/\pi]$
with parameter
$a>0$
.
期刊：
JOURNAL OF INEQUALITIES AND APPLICATIONS,2015年2015(1):1-12 ISSN：1029-242X
通讯作者：
Chu, Yu-Ming
作者机构：
[Chu, Yu-Ming; Song, Ying-Qing; Zhang, Xiao-Hui] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.;[Zhao, Tie-Hong] Hangzhou Normal Univ, Dept Math, Hangzhou 310036, Zhejiang, Peoples R China.
通讯机构：
[Chu, Yu-Ming] H;Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
关键词：
arithmetic mean;Toader mean;quadratic mean
摘要：
In this paper, we present the best possible parameters
$p, q\in\mathbb {R}$
such that the double inequality
$M_{p}(a,b)< T[A(a,b), Q(a,b)]< M_{q}(a,b)$
holds for all
$a, b>0$
with
$a\neq b$
, and we get sharp bounds for the complete elliptic integral
$\mathcal{E}(t)=\int _{0}^{\pi/2}(1-t^{2}\sin^{2}\theta)^{1/2}\,d\theta$
of the second kind on the interval
$(0, \sqrt{2}/2)$
, where
$T(a,b)=\frac{2}{\pi }\int _{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta$
,
$A(a,b)=(a+b)/2$
,
$Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$
,
$M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$
(
$r\neq0$
), and
$M_{0}(a,b)=\sqrt {ab}$
are the Toader, arithmetic, quadratic, and rth power means of a and b, respectively.
期刊：
JOURNAL OF INEQUALITIES AND APPLICATIONS,2015年2015(1):1-13 ISSN：1029-242X
通讯作者：
Chu, Yu-Ming
作者机构：
[Qian, Wei-Mao] Huzhou Broadcast & TV Univ, Sch Distance Educ, Huzhou 313000, Peoples R China.;[Chu, Yu-Ming; Zhang, Xiao-Hui] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
通讯机构：
[Chu, Yu-Ming] H;Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
关键词：
Sándor mean;arithmetic mean;geometric mean;harmonic mean
摘要：
In the article, we present the best possible parameters
$\alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2}\in(0, 1)$
and
$\alpha_{3}, \alpha_{4}, \beta_{3}, \beta_{4}\in(0, 1/2)$
such that the double inequalities
$$\begin{aligned}& \alpha_{1}A(a,b)+(1-\alpha_{1})H(a,b)< X(a,b)< \beta _{1}A(a,b)+(1-\beta_{1})H(a,b), \\ & \alpha_{2}A(a,b)+(1-\alpha_{2})G(a,b)< X(a,b)< \beta _{2}A(a,b)+(1-\beta_{2})G(a,b), \\ & H\bigl[\alpha_{3}a+(1-\alpha_{3})b, \alpha_{3}b+(1- \alpha _{3})a\bigr]< X(a,b)< H\bigl[\beta_{3}a+(1- \beta_{3})b, \beta_{3}b+(1-\beta_{3})a\bigr], \\ & G\bigl[\alpha_{4}a+(1-\alpha_{4})b, \alpha_{4}b+(1- \alpha _{4})a\bigr]< X(a,b)< G\bigl[\beta_{4}a+(1- \beta_{4})b, \beta_{4}b+(1-\beta_{4})a\bigr] \end{aligned}$$
hold for all
$a, b>0$
with
$a\neq b$
. Here,
$X(a,b)$
,
$A(a,b)$
,
$G(a,b)$
and
$H(a,b)$
are the Sándor, arithmetic, geometric and harmonic means of a and b, respectively.
摘要：
For
$x\in R$
, the error function
$\operatorname{erf}(x)$
is defined as
$$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x}e^{-t^{2}}\,dt. $$
In this paper, we answer the question: what are the greatest value p and the least value q, such that the double inequality
$\operatorname {erf}(M_{p}(x,y;\lambda))\leq G(\operatorname{erf}(x),\operatorname {erf}(y);\lambda)\leq\operatorname{erf}(M_{q}(x,y;\lambda))$
holds for all
$x,y\geq1$
(or
$0< x,y<1$
) and
$\lambda\in(0,1)$
? Here,
$M_{r}(x,y;\lambda)=(\lambda x^{r}+(1-\lambda)y^{r})^{1/r}$
(
$r\neq0$
),
$M_{0}(x,y;\lambda)=x^{\lambda}y^{1-\lambda}$
and
$G(x,y;\lambda )=x^{\lambda}y^{1-\lambda}$
are the weighted power and the weighted geometric mean, respectively.
期刊：
JOURNAL OF INEQUALITIES AND APPLICATIONS,2015年2015(1):1-9 ISSN：1029-242X
通讯作者：
Chu, Yu-Ming
作者机构：
[Chu, Yu-Ming; Zhang, Xiao-Hui] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.;[Qian, Wei-Mao] Huzhou Broadcast & TV Univ, Sch Distance Educ, Huzhou 313000, Peoples R China.;[Wu, Li-Min] Huzhou Univ, Dept Math, Huzhou 313000, Peoples R China.
通讯机构：
[Chu, Yu-Ming] H;Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
关键词：
first Seiffert mean;second Seiffert mean;geometric mean;arithmetic mean;contraharmonic mean
摘要：
In this paper, we find the greatest values α, λ and the least values β, μ such that the double inequalities
$\alpha[G(a,b)/3+2A(a,b)/3]+(1-\alpha)G^{1/3}(a,b)A^{2/3}(a,b)< P(a,b) <\beta[G(a,b)/3+2A(a,b)/3]+(1-\beta)G^{1/3}(a,b)A^{2/3}(a,b)$
and
$\lambda [C(a,b)/3+2A(a,b)/3 ]+ (1-\lambda)C^{1/3}(a,b) A^{2/3}(a,b)< T(a,b)<\mu [C(a,b)/3+2A(a,b)/3 ]+(1-\mu)C^{1/3}(a,b) A^{2/3}(a,b)$
hold for all
$a,b>0$
with
$a\neq b$
. Here
$G(a,b)$
,
$A(a,b)$
,
$C(a,b)$
,
$P(a,b)$
and
$T(a,b)$
denote the geometric, arithmetic, contraharmonic, first Seiffert and second Seiffert means of two positive numbers a and b, respectively.
摘要：
We present the necessary and sufficient conditions such that the functions involving
$R ( x ) =\psi ( x+1/2 ) -\ln x$
with a parameter are completely monotonic on
$( 0,\infty )$
, find three new sequences which are fast convergence toward the Euler-Mascheroni constant, and give a positive answer to the conjecture proposed by Chen (J. Math. Inequal. 3(1):79-91, 2009), where ψ is the digamma function.
关键词：
Finite-time global synchronization;Complex network;Markovian jump;Feedback control;Stochastic noises
摘要：
This paper presents a framework for finite-time global synchronization of Markovian jump complex networks (MJCNs) with partially unknown transition rates, time-delays and stochastic noises. Several criteria for finite-time global synchronization are given by using the stochastic analysis techniques and matrix theory. These criteria provide a feasible approach to design linear feedback controller and to compute finite-time of global synchronization for the addressed MJCNs. Four numerical examples are given to demonstrate the effectiveness of the theoretical results.
期刊：
Journal of Function Spaces,2015年2015 ISSN：2314-8896
通讯作者：
Chu, Yu-Ming
作者机构：
[Qian, Wei-Mao] Huzhou Broadcast & TV Univ, Sch Distance Educ, Huzhou 313000, Peoples R China.;[Chu, Yu-Ming; Song, Ying-Qing; Zhang, Xiao-Hui] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
通讯机构：
[Chu, Yu-Ming] H;Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
摘要：
We present the best possible parameters lambda(1), mu(1) is an element of R and lambda(2), mu(2) is an element of(1/2, 1) such that double inequalities lambda(1) C(a,b) + (1 - lambda(1))A(a, b) < T(a, b) < mu C-1(a, b) + (1 - mu(1))A(a, b), C[lambda(2)a + (1 - lambda(2))b, lambda(2)b + (1 - lambda(2))a] < T(a, b) < C[mu(2)a + (1 - mu(2))b, mu(2)b + (1 - mu(2))a] hold for all a, b > 0 with a not equal b, where A(a, b) = (a + b)/2, C(a, b) = (a(3) + b(3))/(a(2) + b(2)) and T(a, b) = 2 integral(pi/2)(0) root a(2)cos(2)theta + b(2)sin(2)theta d theta/pi are the arithmetic, second contraharmonic, and Toader means of a and b, respectively.
期刊：
JOURNAL OF MATHEMATICAL INEQUALITIES,2015年9(3):873-881 ISSN：1846-579X
通讯作者：
Qian, Wei-Mao
作者机构：
[Qian, Wei-Mao] Huzhou Broadcast & TV Univ, Sch Distance Educ, Huzhou 313000, Peoples R China.;[Chu, Yu-Ming] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
通讯机构：
[Qian, Wei-Mao] H;Huzhou Broadcast & TV Univ, Sch Distance Educ, Huzhou 313000, Peoples R China.
关键词：
Schwab-Borchardt mean;Neuman mean;arithmetic mean;contraharmonic mean
摘要：
In this paper, we present the sharp upper and lower bounds for the Neuman means SAC and SCA in terms of the the arithmetic mean A and contraharmonic mean C. The given results are the improvements of some known results.
作者机构：
[Xie, Jin-Shan; Wu, Shan-He] Longyan Univ, Dept Math, Longyan 364012, Fujian, Peoples R China.;[Sroysang, Banyat] Thammasat Univ, Fac Sci & Technol, Dept Math & Stat, Pathum Thani 12121, Thailand.;[Chu, Yu-Ming] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Hunan, Peoples R China.
通讯机构：
[Chu, Yu-Ming] H;Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Hunan, Peoples R China.
关键词：
Hermite-Hadamard type inequality;Parameter;Quasi-convex;Special means
摘要：
In this paper we present some inequalities of Hermite-Hadamard type for functions whose third derivative absolute values are quasi-convex. Moreover, an application to special means of real numbers is also considered.
期刊：
Journal of Function Spaces,2015年2015:1-5 ISSN：2314-8896
通讯作者：
Chu, Yu-Ming
作者机构：
[Chu, Yu-Ming; Song, Ying-Qing] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.;[Wu, Li-Min] Huzhou Univ, Dept Math, Huzhou 313000, Peoples R China.
通讯机构：
[Chu, Yu-Ming] H;Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
摘要：
We present the best possible parameters alpha = alpha(r) and beta = beta(r) such that the double inequality M-alpha(a, b) < H-r(a, b) < M-beta (a, b) holds for all r is an element of ( 0, 1/2) and a, b > 0 with a not equal b where M-p (a, b) = [(a(p) + b(p))/2](1/p) (p not equal 0) and M-0 (a, b) = root ab and H-r (a, b) = 2[ra + (1 - r)b][rb + (1 - r)a]/(a +b) are the power and one- parameter harmonic means of a and b, respectively.
摘要：
In this paper, state estimation for discrete-time BAM neural networks with time-varying delay is discussed. Under a weaker assumption on activation functions, by constructing a novel Lyapunov-Krasovskii functional (LKF), a set of sufficient conditions are derived in terms of linear matrix inequality (LMI) for the existence of state estimator such that the error system is global exponentially stable. Based on the delay partitioning method and the reciprocally convex approach, some less conservative stability criteria along with lower computational complexity are obtained. Finally, a numerical example is given to show the effectiveness of the derived result.
期刊：
JOURNAL OF INEQUALITIES AND APPLICATIONS,2015年2015(1):1-10 ISSN：1029-242X
通讯作者：
Chu, Yu-Ming
作者机构：
[Zhao, Tie-Hong] Hangzhou Normal Univ, Dept Math, Hangzhou 310036, Zhejiang, Peoples R China.;[Yang, Zhen-Hang] Zhejiang Elect Power Corp, Power Supply Serv Ctr, Res Inst, Hangzhou 310007, Zhejiang, Peoples R China.;[Chu, Yu-Ming] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
通讯机构：
[Chu, Yu-Ming] H;Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
关键词：
gamma function;psi function;monotonicity
摘要：
In this paper, we present the best possible parameter
$a\in(1/15, \infty)$
such that the functions
$\psi^{\prime}(x+1)-\mathcal{L}_{x}(x, a)$
and
$\psi^{\prime\prime }(x+1)-\mathcal{L}_{xx}(x, a)$
are strictly increasing or decreasing with respect to
$x\in(0, \infty)$
, where
$\mathcal{L}(x,a)=\frac {1}{90a^{2}+2}\log (x^{2}+x+\frac{3a+1}{3} )+ \frac {45a^{2}}{90a^{2}+2}\log (x^{2}+x+\frac{15a-1}{45a} )$
and
$\psi(x)$
is the classical psi function. As applications, we get several new sharp bounds for the psi function and its derivatives.
期刊：
Applied Mathematics and Computation,2015年268:1055-1063 ISSN：0096-3003
通讯作者：
Chu, Yu-Ming
作者机构：
[Chu, Yu-Ming; Yang, Zhen-Hang; Zhang, Xiao-Hui] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
通讯机构：
[Chu, Yu-Ming] H;Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
关键词：
Gamma function;Psi function;Monotonicity
摘要：
In this paper, we present the greatest value a is an element of (1/15, infinity) and the least value b is an element of (4/15, infinity) such that the double inequality L(x, a) < psi/ (x + 1) < L(x - 1, b) +1/x holds for all x > 0, where L(x, a) = 190a(2)+2 log (x(2) + x + 3a+1/3) + 45a(2)/90a(2)+2 log (x(2) + x + 15a-1/45a) and psi(x) is the classical psi function. (C) 2015 Elsevier Inc. All rights reserved.
作者机构：
[Wang MiaoKun; Chu YuMing] Hunan Univ, Coll Math & Econometr, Changsha 410082, Hunan, Peoples R China.;[Song YingQing] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
通讯机构：
[Wang MiaoKun] H;Hunan Univ, Coll Math & Econometr, Changsha 410082, Hunan, Peoples R China.
关键词：
Gaussian hypergeometric function;Ramanujan's cubic transformation;generalized modular equation;infinite product;modular function
摘要：
We study the quotient of hypergeometric functionsμa*（r）=π/2sin（πa） F（a,1-a;1;1-r3）/F（a,1-a;1;r3）,r∈（0,1）in the theory of Ramanujan’s generalized modular equation for a ∈(0,1/2],and find an infinite product formula for μ1/3*（r） by use of the properties of μa*（r） and Ramanujan’s cubic transformation.Besides,a new cubic transformation formula of hypergeometric function is given,which complements the Ramanujan’s cubic transformation.
期刊：
JOURNAL OF MATHEMATICAL INEQUALITIES,2015年9(3):699-707 ISSN：1846-579X
通讯作者：
He, Zai-Yin
作者机构：
[Chu, Yu-Ming; Wang, Miao-Kun; He, Zai-Yin] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
通讯机构：
[He, Zai-Yin] H;Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China.
关键词：
Arithmetic mean;Logarithmic mean;One-parameter mean
摘要：
We find the greatest value p(1) = p(1)(alpha) and the least value p(2) = p(2)(alpha) such that the double inequality J(p1) (a, b) < alpha A(a, b)+(1-alpha)L(a, b) < J(p2) (a, b) holds for any alpha is an element of (0,1) and all a, b > 0 with a not equal b. Here, A(a, b), L(a, b) and J(p)(a, b) denote the arithmetic, logarithmic and p-th one-parameter means of two positive numbers a and b, respectively.
