Conditions for the representability of analytic functions by series of rational functions

With a functionf(z), analytic in the unit circle, we associate by a specific rule the series \(\sum\nolimits_{n = 1}^\infty {\frac{{A_n }}{{1 - \lambda _n z}},\left| {\lambda _n } \right|< 1} \) . we derive a (necessary and sufficient) condition for the convergence of the series in the unit circle. We derive further conditions under which the series converges to the functionf(z) itself.     

On the Li-Stevi? Integral Type Operators from Weighted Bergman Spaces into β-Zygmund Spaces

For an analytic self-map ?? of the unit disk ${\mathbb{D}}$ and an analytic function g on ${\mathbb{D}}$ , we define the following integral type operators: $$T_{\varphi}^{g}f(z) := \int_{0}^{z} f(\varphi(\zeta))g(\zeta) d\zeta\quad {\rm and}\quad C_{\varphi}^{g}f(z) := \int_{0}^{z}f^{\prime}(\varphi(\zeta))g(\zeta) d\zeta$$ . We give a characterization for the boundedness and compactness of these operators from the weighted Bergman space ${L_{a}^p(dA_{\alpha})}$ into the ??-Zygmund space ${\mathcal{Z}_{\beta}}$ . We will also estimate the essential norm of these type of operators. As an application of results, we characterize the above operator-theoretic properties of Volterra type integral operators and composition operators.     

More quantitative results on walsh equiconvergence: I. Lagrange Case

Letf∈A _ρ (ρ>1), whereA _ρ denotes the class of functions analytic in ¦z¦ <ρ but not in ¦z¦≤ρ. For any positive integerl, the quantity Δ _l,n?1(f; z) (see (2.3)) has been studied extensively. Recently, V. Totik has obtained some quantitative estimates for \(\overline {\lim _{n \to \infty } } \max _{\left| z \right| = R} \left| {\Delta _{l,n - 1}^ - \left( {f;z} \right)} \right|^{1/n} \) . Here we investigate the order of pointwise convergence (or divergence) of Δ _l,n?1(f; z), i.e., we study \(B_1 \left( {f;z} \right) = \overline {\lim _{n \to \infty } } \left| {\Delta _{l,n - 1} \left( {f;z} \right)} \right|^{1/n} \) . We also study some problems arising from the results of Totik.     

Coefficient inequalities for univalent starlike functions

Let A be the class of analytic functions in the unit disk $\mathbb{D}$ with the normalization f(0) = f′(0) ? 1 = 0. In this paper the authors discuss necessary and sufficient coefficient conditions for f ∈ A of the form $$\left( {\frac{z} {{f(z)}}} \right)^\mu = 1 + b_1 z + b_2 z^2 + \ldots$$ to be starlike in $\mathbb{D}$ and more generally, starlike of some order β, 0 ≤ β < 1. Here µ is a suitable complex number so that the right hand side expression is analytic in $\mathbb{D}$ and the power is chosen to be the principal power. A similar problem for the class of convex functions of order β is open.     

Decomposition theorems and conjugate pair in D K spaces

Let K be a right-continuous and nondecreasing function.A function f analytic in the unit disk D belongs to the space DK if D|f(z)|2K(1- |z|2)dA(z) ∞.Decomposition theorems for DK spaces are established in this paper.As an application,we obtain a characterization of interpolation by functions in DKspaces.Furthermore,we characterize functions in DKspaces by conjugate pairs.     

Estimates for the coefficients of univalent functions in terms of the second coefficient

For the coefficients b_n of an odd function \(f(z) = z + \sum\nolimits_{k = 1}^\infty {{}^bk^{z^{2k + 1} } } \) , regular in the unit disk, we obtain the estimate $$|b_n | \leqslant \frac{1}{{\sqrt 2 }}\sqrt {1 + |b_1 |^2 } \exp \frac{1}{2}\left( {\delta + \frac{1}{2}|b_1 |^2 } \right),where \delta = 0.312,$$ (1) from which it follows that ¦b_n¦≤1, if ¦b₁¦≤0.524. It follows from (1) that the coefficients c_n, n = 3, 4,..., of a regular function \(f(2) = z + \sum\nolimits_{k = 2}^\infty {{}^ck^{z^k } } \) , univalent in the unit desk, satisfy $$|c_n | \leqslant \frac{1}{2}\left( {1 + \frac{{|c_2 |^2 }}{4}} \right)n\exp \left( {\delta + \frac{{|c_2 |^2 }}{8}} \right),where \delta = 0.312,$$ in particular, ¦c_n¦≤n, if ¦c₂¦≤1.046.     

Banach spaces of locally Schlicht functions with the Hornich operations

Let \(S_ \propto ( \propto \geqq 0)\) be the set of normalized (see (1.2)) functions f holomorphic in D:|z|<1 with \(f''(z)/f'(z) = 0((1 - \left| z \right|^2 )^{ - \propto } )\) , and let be the set of normalized (see (1.6)) functions f meromorphic in D with the Schwarzian derivative \(\left\{ {f,z} \right\} = 0((1 - \left| z \right|^2 )^{ - \propto } )\) . We shall show that some topological properties of \(S_ \propto\) and , and of subsets of them, follow from those of the weighted H^∞ space \(H_ \propto ^\infty\) , consisting of functions f holomorphic in D with \(f(z) = 0((1 - \left| z \right|^2 )^{ - \propto } )\) , and those of subsets of \(H_ \propto ^\infty\) . The set S₁ is denoted by X in [3] and [4].     

Partial regularity for non autonomous functionals with non standard growth conditions

We prove a C ^1,μ partial regularity result for minimizers of a non autonomous integral funcitional of the form $$\mathcal{F}(u; \Omega):=\int_{\Omega}f(x, Du)\ dx$$ under the so-called non standard growth conditions. More precisely we assume that $$c |z|^{p}\leq f(x ,z) \leq L (1+|z|^{q}),$$ for 2 ≤ p < q and that D _z f(x, z) is α-Hölder continuous with respect to the x-variable. The regularity is obtained imposing that ${\frac{p}{q} < \frac{n+\alpha}{n}}$ but without any assumption on the growth of ${D^{2}_{z}f}$ .     

Sharp weak-type inequalities for Hilbert transform and Riesz projection

The paper is devoted to the study of the weak norms of the classical operators in the vector-valued setting.

1. Let S, H denote the singular integral involution operator and the Hilbert transform on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {\mathcal{S}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p,$$ $$\left\| {\mathcal{H}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p.$$ Both inequalities are sharp.
2. Let P ₊ and P _? stand for the Riesz projection and the co-analytic projection on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {P + f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p ,$$ $$\left\| {P - f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p .$$ Both inequalities are sharp.
3. We establish the sharp versions of the estimates above in the nonperiodic case.

The results are new even if the operators act on complex-valued functions. The proof rests on the construction of an appropriate plurisubharmonic function and probabilistic techniques.     

On the solutions of a singular elliptic equation concentrating on two orthogonal spheres

Let \({A=\{x\in \mathbb{R}^{2m}: 0 < a < |x| < b\}}\) be an annulus. We consider the following singularly perturbed elliptic problem on A $$\left\{\begin{array}{lll}-\varepsilon ^2{\Delta u} + |x|^{\eta}u =|x|^{\eta}u^p, \quad {\rm in} A,\\ u > 0, \quad \quad \quad \quad \quad \quad \quad {\rm in} A, \\ u=0, \quad \quad \quad \quad \quad \quad \quad {\rm on}\partial A,\end{array}\right. $$ where \({1 < p < \frac{m+3}{m-1}}\) . We shall prove the existence of a positive solution \({u_\epsilon }\) which concentrates on two different orthogonal spheres of dimension (m?1) as \({\varepsilon \to 0}\) . We achieve this by studying a reduced problem on an annular domain in \({\mathbb{R}^{m+1}}\) and analysing the profile of a two point concentrating solution in this domain.     

Bergman type operators on a class of weakly pseudoconvex domains

A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on L^p (\Omega _a ,dv_\lambda ),1< p< \infty $ , is obtained, where $\Omega _a = \left\{ {\xi = (z,w) \in \mathbb{C}^{n + m} :z \in \mathbb{C}^n ,w \in \mathbb{C}^m ,|z|^2 + |w|^{2/a}< 1} \right\},h(\xi ) = (1 - |z|^2 )^a - |w|^2 $ andK _x,u,v (ξ,ξ′).This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain ω _a . As an appli cation, it is proved thatf?L _H ^p (ω _a ,dv _λ) implies $h\tfrac{{|a|}}{a} + |\beta |(\xi )D_2^a D_z^\beta f \in L^p (\Omega _a ,dv_\lambda ),1 \leqslant p< \infty $ , for any multi-indexa=(α₁,?,α _n and ß = (ß₁, —ß). An interesting question is whether the converse holds.     

Schwache Lösungen des Frankl-Morawetz-Problems im ℝ3

In a bounded simple connected region G ? ?³ we consider the equation $$L\left[ u \right]: = k\left( z \right)\left( {u_{xx} + u_{yy} } \right) + u_{zz} + d\left( {x,y,z} \right)u = f\left( {x,y,z} \right)$$ where k(z)? 0 whenever z ? 0.G is surrounded forz≥0 by a smooth surface Γ₀ with S:=Γ₀ ? {(x,y,z)|=0} and forz<0 by the characteristic \(\Gamma _2 :---(x^2 + y^2 )^{{\textstyle{1 \over 2}}} + \int\limits_z^0 {(---k(t))^{{\textstyle{1 \over 2}}} dt = 0} \) and a smooth surface Γ₁ which intersect the planez=0 inS and where the outer normal n=(n_x, n_y, n_z) fulfills \(k(z)(n_x^2 + n_y^2 ) + n_z^2 |_{\Gamma _1 } > 0\) . Under conditions on Γ₁ and the coefficientsk(z), d(x,y,z) we prove the existence of weak solutions for the boundary value problemL[u]=f inG with \(u|_{\Gamma _0 \cup \Gamma _1 } = 0\) . The uniqueness of the classical solution for this problem was proved in [1].     

Semiclassical limits of ground state solutions to Schrödinger systems

This paper is concerned with the existence and concentration properties of the ground state solutions to the following coupled Schrödinger systems $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)G_{v}(z)~\hbox { in }\ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)G_{u}(z)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ and $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)(G_{v}(z)+|z|^{2^*-2}v)~\hbox {in } \ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)(G_{u}(z)+|z|^{2^*-2}u)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where \(z=(u,v)\in {\mathbb {R}}^2\) , \(G\) is a power type nonlinearity, having superquadratic growth at both \(0\) and infinity but subcritical, \(V\) can be sign-changing and \(\inf W>0\) . We prove the existence, exponential decay, \(H^2\) -convergence and concentration phenomena of the ground state solutions for small \(\varepsilon >0\) .     

Boundary parametrization of planar self-affine tiles with collinear digit set

We consider a class of planar self-affine tiles T = M-1 a∈D(T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:M =(0-B 1-A),D = {(00),...,(|B|0-1)}.We give a parametrization S1 →T of the boundary of T with the following standard properties.It is H¨older continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on T and have algebraic preimages.We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.     

О РАцИОНАльНых сОстА ВльУЩИх МЕРОМОРФНых ФУНкцИИ И Их пРОИжВОД Ных

LetG be an arbitrary domain in \(\bar C\) ,f a function meromorphic inG, $$M_f \mathop = \limits^{def} \mathop {\lim \sup }\limits_{G \mathrel\backepsilon z \to \partial G} \left| {f(z)} \right|< \infty ,$$ andR the sum of the principal parts in the Laurent expansions off with respect to all its poles inG. We set $$f_G (z) = R(z) - \alpha ,{\mathbf{ }}where{\mathbf{ }}\alpha = \mathop {\lim }\limits_{z \to \infty } (f(z) - R(z))$$ in case ∞?G, andα=0 in case ∞?G. It is proved that $$\left\| {f_G } \right\|_{C(\partial G)} \leqq 50(\deg f_G )M_f ,{\mathbf{ }}\left\| {f'_G } \right\|_{L_1 (\partial G)} \leqq 50(\deg f_G )V(\partial G)M_f ,$$ where $$V(\partial G) = \sup \left\{ {\left\| {r'} \right\|_{L_1 (\partial G)} :r(z) = a/(z - b),{\mathbf{ }}\left\| r \right\|_{G(\partial G)} \leqq 1} \right\}.$$      

Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk ${\mathbb{D}}$ . Each function ${f\in Co(\alpha)}$ maps the unit disk ${\mathbb{D}}$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional ${(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ . In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional ${(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

One-radius results for supermedian functions on {\mathbb R^d} , d ≤ 2

A classical result states that every lower bounded superharmonic function on ${\mathbb{R}^{2}}$ is constant. In this paper the following (stronger) one-circle version is proven. If ${f : \mathbb{R}^{2} \to (-\infty,\infty]}$ is lower semicontinuous, lim inf_|x|→∞ f (x)/ ln |x| ≥ 0, and, for every ${x \in \mathbb{R}^{2}}$ , ${1/(2\pi) \int_0^{2\pi} f(x + r(x)e^{it}) \, dt \le f(x)}$ , where ${r : \mathbb{R}^{2} \to (0,\infty)}$ is continuous, ${{\rm sup}_{x \in \mathbb{R}^{2}} (r(x) - |x|) < \infty},$ , and ${{\rm inf}_{x \in \mathbb{R}^{2}} (r(x)-|x|)=-\infty}$ , then f is constant. Moreover, it is shown that, assuming r ≤ c| · | + M on ${\mathbb{R}^d}$ , d ≤ 2, and taking averages on ${\{y \in \mathbb{R}^{d} : |y-x| \le r(x)\}}$ , such a result of Liouville type holds for supermedian functions if and only if c ≤ c ₀, where c ₀ = 1, if d = 2, whereas 2.50 < c ₀ < 2.51, if d = 1.     

Some new estimates of the Fourier transform in \mathbb{L}_2 (ℝ)

Given a function $\mathbb{L}_2 $ (?), its Fourier transform $g(x) = \hat f(x) = F[f](x) = \frac{1} {{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {f(x)e^{ - ixt} dt} ,f(t) = F^{ - 1} [g](t) = \frac{1} {{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {g(x)e^{ - ixt} dx} $ and the inverse Fourier transform are considered in the space f ε $\mathbb{L}_2 $ (?). New estimates are presented for the integral $\int\limits_{|t| \geqslant N} {|g(t)|^2 dt} = \int\limits_{|t| \geqslant N} {|\hat f(t)|^2 dt} ,N \geqslant 1,$ in the vase of f ε $\mathbb{L}_2 $ (?) characterized by the generalized modulus of continuity of the kth order constructed with the help of the Steklov function. Some other estimates associated with this integral are proved.