Positive Gegenbauer Polynomial Sums and Applications to Starlike Functions

Let $s_n(f,z):=\sum_{k=0}^{n}a_kz^k$ be the $n$th partial sum of $f(z)=\sum_{k=0}^{\infty{}}a_kz^k$. We show that $\RE s_n(f/z,z)>0$ holds for all $z\in\D,\ n\in\N$, and all starlike functions $f$ of order $\lambda$ iff $\lambda_0\leq\lambda<1$ where $\lambda_0=0.654222...$ is the unique solution $\lambda\in(\frac{1}{2},1)$ of the equation $\int_{0}^{3\pi/2}t^{1-2\lambda}\cos t \,dt=0$. Here $\D$ denotes the unit disk in the complex plane $\C$. This result is the best possible with respect to $\lambda_0$. In particular, it shows that for the Gegenbauer polynomials $C_{n}^{\mu}(x)$ we have $\sum_{k=0}^n C_{k}^{\mu}(x)\cos k \theta>0$ for all $n\in\N,\ x\in[-1,1]$, and $0<\mu\leq\mu_0:=1-\lambda_0=0.345778...$. This result complements an inequality of Brown, Wang, and Wilson (1993) and extends a result of Ruscheweyh and Salinas (2000).     

Nodal Solutions of the Brezis-Nirenberg Problem in Dimension 6

We show that the classical Brezis-Nirenberg problem $$-\Delta u=u|u|+\lambda u \ \ \ \ \ \ \ in \ \ \ \Omega, \\ u=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ on \ \ \ \partial\Omega,$$ when $\Omega$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $\Omega$ as $\lambda$ approaches a suitable value $\lambda_0>0.$     

An Existence Result for a Class of Chemically Reacting Systems with Sign-Changing Weights

We prove the existence of positive solutions for the system$$\begin{align*}\begin{cases}-\Delta_{p} u =\lambda a(x){f(v)}{u^{-\alpha}},\qquad x\in \Omega,\\-\Delta_{q} v = \lambda b(x){g(u)}{v^{-\beta}},\qquad x\in \Omega,\\u = v =0, \qquad x\in\partial \Omega,\end{cases}\end{align*}$$where $\Delta_{r}z={\rm div}(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator and $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\alpha, \beta \in (0,1).$ Here $ a(x)$ and $ b(x)$ are $C^{1}$ sign-changingfunctions that maybe negative near the boundary and $f,g $ are $C^{1}$ nondecreasing functions, such that $f, g :\ [0,\infty)\to [0,\infty);$ $f(s)>0,$ $g(s)>0$ for $s> 0$, $\lim_{s\to\infty}g(s)=\infty$ and$$\lim_{s\to\infty}\frac{f(Mg(s)^{\frac{1}{q-1}})}{s^{p-1+\alpha}}=0,\qquad \forall M>0.$$We discuss the existence of positive weak solutions when $f$, $g$, $a(x)$ and $b(x)$ satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.     

二项式展开的抵消系数以及$q$-摸拟

Let {An}∞n=0 be an arbitary sequence of natural numbers. We say A(n,k;A) are the Convolution Annihilation Coefficients for {An}n∞=0 if and only if n k=0 A(n,k;A)(x - Ak)n-k = xn. (0.1) Similary, we define B(n,k;A) to be the Dot Product Annihilation Coefficients for {An}n∞=0 if and only if n k=0 B(n,k;A)(x - Ak)k = xn. (0.2) The main result of this paper is an explicit formula for B(n,k;A), which depends on both k and {An}∞n=0. This paper also discusses binomial and q-analogs of Equations (0.1) and (0.2).     

On a class of singular p-Laplacian semipositone problems with sign-changing weights

We study existence of positive weak solution for a class of $p$-Laplacian problem $$\left\{\begin{array}{ll}-\Delta_{p}u = \lambda g(x)[f(u)-\frac{1}{u^{\alpha}}], &amp; x\in \Omega,\\u= 0 , &amp; x\in\partial \Omega,\end{array\right.$$ where $\lambda$ is a positive parameter and $\alpha\in(0,1),$ $\Omega $ is a bounded domain in $ R^{N}$ for $(N &gt; 1)$ with smooth boundary, $\Delta_{p}u = div (|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian operator for $( p &gt; 2),$ $g(x)$ is $C^{1}$ sign-changing function such that maybe negative near the boundary and be positive in the interior and $f$ is $C^{1}$ nondecreasing function $\lim_{s\to\infty}\frac{f(s)}{s^{p-1}}=0.$ We discuss the existence of positive weak solution when $f$ and $g$ satisfy certain additional conditions. We use the method of sub-supersolution to establish our result.     

A Connection Between the Logarithmic Capacity and the Sequence of the Analytical Function

Let E be a bounded closed set, d(E) be the logarithmic capacity of E. If A is any bounded set, then $[d(A) = \mathop {\sup }\limits_{E \in A} d(E)\]$ For each $Z_0 \in E$, and $\delta >0$, let $[\Delta = \Delta _{{Z_0}}^\delta = CE \cap (|Z - {Z_0}| < \delta )\]$ where CE is complement of E, then \Delta is an open set. By [{\bar \Delta ^0}\] we denote the interior of the closure A of A. Clearly,$\Delta \subset [{\bar \Delta ^0}\]$ and $d(\Delta) \leq d([{\bar \Delta ^0}\])$, and there exists an open set D such that d(D) 0, the equation $d(\Delta)=d([{\bar \Delta ^0}\])$ holds.     

Transplantation Theorems for Ultraspherical Polynomials in Re H 1 and BMO

We consider the uniformly bounded orthonormal system of functions $$ u_n^{(\l)}(x)= \varphi_n^{(\lambda)}(\cos x)(\sin x)^\lambda, \qquad x\in [0,\pi], $$ where $\{\varphi_n^{(\lambda)}\}_{n=0}^\infty \,\, (\lambda > 0)$ is the normalized system of ultraspherical polynomials. R. Askey and S. Wainger proved that the $L^p$-norm $(1 < p < \infty)$ of any linear combination of the first $N+1$ functions $u_n^{(\lambda)}(x)$ is equivalent to the $L^p$-norm of the even trigonometric polynomial of degree $N$ with the same coefficients. This theorem fails if $p=1 $ or $p=\infty.$ Studying these limiting cases, we prove (for $0 < \lambda < 1$) similar transplantation theorems in $\mbox{Re } H^1$ and $\mbox{BMO}.$     

Local Explicit Many-Knot Spline Hermite Approximation Schemes

If $f^{(i))}(\alpha)(\alpha=a, i=0,1,...,k-2)$ are given, then we get a class of the Hermite approximation operator Qf=F satisfying $F^{(i)}(\alpha)=f^{(i)}(\alpha)$, where F is the many-knot spline function whose knots are at points $y_i:$=$y_0$<$y_1$<$\cdots$<$y_{k-1}=b$, and $F\in P_k$ on $[y_{i-1},y_i]$. The operator is of the form $Qf:=\sum\limits_{i=0}^{k-2}[f^{(i)}(a)\phi_i+f^{(i)}(b)\psi _i]$. We give an explicit representation of $\phi_i$ and $\psi_i$ in terms of B-splines $N_{i,k}$. We show that Q reproduces appropriate classes of polynomials.     

The Integer Hull of a Convex Rational Polytope

Given $A\in\Z^{m\times n}$ and $b\in\Z^m$, we consider the integer program $\max \{cx\vert Ax=b;x\in\N^n\}$ and provide an {\it equivalent} and {\it explicit} linear program $\max \{\widehat{\xcc}q\vert \m q=r;q\geq 0\}$, where $\m,r,\widehat{c}$ are easily obtained from $A,b,c$ with no calculation. We also provide an explicit algebraic characterization of the integer hull of the convex polytope $\p=\{x\in\R^n\vert Ax=b;x\geq0\}$. All strong valid inequalities can be obtained from the generators of a convex cone whose definition is explicit in terms of $\m$.     

Boundary value problem for the Burgers system

We consider the boundary value problem $\begin{gathered} div(\rho V) = 0, \rho |\Gamma _1 = \rho 0, \hfill \\ \rho (V,\nabla V) = v\Delta V, V|\Gamma = V^0 \hfill \\ \end{gathered} $ for a vector functionV=(V ₁,V ₂) and a scalar function ρ>-0 in a rectangular domain Ω ? ?² with boundary Γ. Here $\Gamma _1 = \{ x \in \Gamma :(V^0 ,n)< 0\} , V_1^0 |_\Gamma > 0, v = const > 0.$ We prove that this problem is solvable in Hölder classes.     

The behaviour of microstructures with small shears of the austenite-martensite interface in martensitic phase transformations

Let $\Omega \subset \Bbb{R}^2$ denote a bounded domain whose boundary $\partial \Omega$ is Lipschitz and contains a segment $\Gamma_0$ representing the austenite-twinned martensite interface. We prove $$\displaystyle{\inf_{{u\in \cal W}(\Omega)} \int_\Omega \varphi(\nabla u(x,y))dxdy=0}$$ for any elastic energy density $\varphi : \Bbb{R}^2 \rightarrow [0,\infty)$ such that $\varphi(0,\pm 1)=0$. Here ${\cal W}(\Omega)$ consists of all Lipschitz functions $u$ with $u=0$ on $\Gamma_0$ and $|u_y|=1$ a.e. Apart from the trivial case $\Gamma_0 \subset \reel \times \{a\},~a\in \Bbb{R}$, this result is obtained through the construction of suitable minimizing sequences which differ substantially for vertical and non-vertical segments.     

Rate Optimality of Wavelet Series Approximations of Fractional Brownian Motion

Consider the fractional Brownian motion process $B_H(t), t\in [0,T]$, with parameter $H\in (0,1)$. Meyer, Sellan and Taqqu have developed several random wavelet representations for $B_H(t)$, of the form $\sum_{k=0}^\infty U_k(t)\epsilon_k$ where $\epsilon_k$ are Gaussian random variables and where the functions $U_k$ are not random. Based on the results of Kühn and Linde, we say that the approximation $\sum_{k=0}^n U_k(t)\epsilon_k$ of $B_H(t)$ is optimal if $$ \displaystyle \left( E \sup_{t\in [0,T]} \left| \sum_{k=n}^\infty U_k(t) \epsilon_k\right|^2 \right)^{1/2} =O \left( n^{-H} (1+\log n)^{1/2} \right), $$ as $n\rightarrow\infty$. We show that the random wavelet representations given in Meyer, Sellan and Taqqu are optimal.     

次线性条件下奇异三阶微分方程周期边值问题解的全局结构

讨论三阶微分方程周期边值问题解的全局结构,其中ρ∈(0,1/3~(1/2))为常数,λ∈R~+=[0,+∞)为参数,f在t=0,t=2π和u=0处有奇异性,关于u处满足次线性增长条件。     

A Singular Parabolic Boundary Value Problem in Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

We consider the operator ${\cal A}$ formally defined by ${\cal A}u(x)=\alpha(x)\Delta u(x)$ for any $x$ in a sufficiently smooth bounded open set $\Om\subset\R^N$ ($N\ge 1$), where $\alpha\in C(\ov\Omega)$ is a continuous nonnegative function vanishing only on $\partial\Omega$, and such that $1/\alpha$ is integrable in $\Omega$. We prove that the realization $A_p$ of ${\cal A}$, equipped with suitable nonlinear boundary conditions is an m-dissipative operator in suitably weighted $L^p(\Omega)$-spaces in the case where either $(p,N)\in (1,+\infty)\times\{1\}$ or $(p,N)=\{2\}\times\N$. Moreover, we prove that $A_p$ is a densely defined closed operator. We consider nonlinear boundary conditions of the following type: in the one dimensional case we take $\Omega=(0,1)$ and we assume that $u(j)=(-1)^j\beta_j(u(j))$ ($j=0,1$), $\beta_0$ and $\beta_1$ being nondecreasing continuous functions in $\R$ such that $\beta_0(0)=\beta_1(0)=0$; in the $N$-dimensional setting we assume that $(D_{\nu}u)_{|\partial\Omega}=-\beta(u_{|\partial\Omega})$, $\beta$ being a nondecreasing Lipschitz continuous function in $\R$ such that $\beta(0)=0$. Here $\nu$ denotes the unit outward normal to $\partial\Om$.     

关于广义{\Phi}-增生算子的最速下降法的收敛定理

设$E$为一致光滑Banach空间,$A:E\to E$为有界次连续广义${\it \Phi} $-增生算子满足:对任意$x_0\in E$,选取$m\ge 1$,使得$\| x_0 - x^* \| \le m$且$\mathop {\underline {\lim } }\limits_{r \to \infty } {\it \Phi} (r) > m\left\| {Ax_0 } \right\|$.设$\{C_n\}$为$[0,1]$中数列满足控制条件: i)$C_n\to 0\,(n\to\infty)$; ii)$\sum\limits_{n = 0}^\infty {C_n } = \infty $.设$\{x_n\}_{n\ge0}$由下式产生x_{n + 1} = x_n - C_n Ax_n ,\q n \ge 0, \eqno{(@)}$$则存在常数$a>0$,当$C_n < a$时,$\{x_n\}$强收敛于$A$的唯一零点$x^{*}$.     

The Krzyz Problem and Polynomials with Zeros on the Unit Circle

Let $ \Pi_{n,M} $ be the class of all polynomials $ p(z) = \sum _{0}^{n} a_{k}z^{k} $ of degree n which have all their zeros on the unit circle $ |z| = 1$ , and satisfy $ M = \max _{|z| = 1}|\,p(z)| $ . Let $ \mu _{k,n} = \sup _{p\in \Pi _{n,M}} |a_{k}| $ . Saff and Sheil-Small asked for the value of $\overline {\lim }_{n\rightarrow \infty }\mu _{k,n} $ . We find an equivalence between this problem and the Krzyz problem on the coefficients of bounded non-vanishing functions. As a result we compute $$ \overline {\lim }_{n\rightarrow \infty }\mu _{k,n} = {{M} \over {e}}\quad {\rm for}\ k = 1,2,3,4,5.$$ We also obtain some bounds for polynomials with zeros on the unit circle. These are related to a problem of Hayman.     

Limit Boundary Value Problems of a class of Nonlinear Differential Equations of the Second order

In this paper, the existence and uniqueness of solution of the limit boundary value problem $\[\ddot x = f(t,x)g(\dot x)\]$(F) $\[a\dot x(0) + bx(0) = c\]$(A) $\[x( + \infty ) = 0\]$(B) is considered, where $\[f(t,x),g(\dot x)\]$ are continuous functions on $\[\{ t \ge 0, - \infty < x,\dot x < + \infty \} \]$ such that the uniqueness of solution together with thier continuous dependence on initial value are ensured, and assume: 1)$\[f(t,0) \equiv 0,f(t,x)/x > 0(x \ne 0);\]$; 2) f(t,x)/x is nondecreasing in x>0 for fixed t and non-increasing in x<0 for fixed t, 3)$\[g(\dot x) > 0\]$, In theorem 1, farther assume: 4) $\[\int\limits_0^{ \pm \infty } {dy/g(y) = \pm \infty } \]$ Condition (A) may be discussed in the following three cases $x(0)=p(p \neq 0)$(A_1) $\[x(0) = q(q \ne 0)\]$(A_2) $\[x(0) = kx(0) + r{\rm{ }}(k > 0,r \ne 0)\]$(A_3) The notation $\[f(t,x) \in {I_\infty }\]$ will refer to the function f(t,x) satisfying $\[\int_0^{ + \infty } {\alpha tf(t,\alpha )dt = + \infty } \]$ for each $\alpha \neq 0$, Theorem. 1. For each $p \neq 0$, the boundary value problem (F), (A_1), (B) has a solution if and only if $f(t,x) \in I_{\infty}$ Theorem 2. For each$q \neq 0$, the boundary value problem (F), (A_2), (B) has a solution if and only if $f(t, x) \in I_{\infty}$. Theorem 3. For each k>0 and $r \neq 0$, the boundary value problem (F), (A_3), (B) has a solution if and only if f(t, x) \in I_{\infty}, Theorem 4. The boundary value problem (F), (A_j), (B) has at most one solution for j=l, 2, 3. .     

UNUNIQUENESS OF SOME HOLOMORPHIC FUNCTIONS

In the present paper, we show that there exist a bounded, holomorphic function $\[f(z) \ne 0\]$ in the domain $\[\{ z = x + iy:\left| y \right| < \alpha \} \]$ such that $\[f(z)\]$ has a Dirichlet expansion $\[\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ in the halfplane $\[x > {x_f}\]$ if and only if $\[\frac{a}{\pi }\log r - \sum\limits_{{u_n} < r} {\frac{2}{{{u_n}}}} \]$ has a finite upperbound on $\[[1, + \infty )\]$, where $\[\alpha \]$ is a positive constant,$\[{x_f}( < + \infty )\]$ is the abscissa of convergence of $\[\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ and the infinite sequence $\[\{ {u_n}\} \]$ satisfies $\[\mathop {\lim }\limits_{n \to + \infty } ({u_{n + 1}} - {u_n}) > 0\]$. We also point out some necessary conditions and sufficient ones Such that a bounded holomorphic function in an angular(or half-band) domain is identically zero if an infinite sequence of its derivatives and itself vanish at some point of the domain. Here some result are generalizations of those in [4].     

Uniqueness and existence of solutions for a singular system with nonlocal operator via perturbation method

In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows: $$ \left\{\begin{array}{ll} (-\Delta_p)^su = a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\ (-\Delta_p)^s v= b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega,\ u=v = 0 ,\;\;\mbox{ in }\,\mathbb{R}^N\setminus\Omega, \end{array} \right. $$ where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary, $0<\alpha <1,$ $0<\beta <1,$ $2-\alpha -\beta 相似文献   

EXISTENCE AND NON-EXISTENCE OF GLOBAL SMOOTH SOLUTIONS FOR QUASILINEAR HYPERBOLIC SYSTEMS~*

Consider initial value probiom v_t-u_x=0, u_t+p(v)_x=0, (E), v(x, 0)=v_0(x), u(x, 0)=u_0(x), (I), where A≥0, p(v)=K~2v~(-γ), K>0, 0<γ<3. As 0<γ≤1, the authors give a sufficient condition for that (E), (I) to have a unique global smooth solution, As 1≤γ<3, a necessary condition is given for that.