On Classes of Functions Related to Starlike Functions with Respect to Symmetric Conjugate Points Defined by a Fractional Differential Operator

Let A denote the class of analytic functions f, in the open unit disk E = {z : |z| < 1}, normalized by f(0) = f′(0) − 1 = 0. In this paper, we introduce and study the class ST^n,a_l,m(h){ST^{n,\alpha}_{\lambda,m}(h)} of functions f ? A{f\in A}, with \fracD^n,a_l f_m(z)z 1 0{\frac{D^{n,\alpha}_\lambda f_m(z)}{z}\neq 0}, satisfying

On Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

A Quantitative Version of the Non-Abelian Idempotent Theorem

Suppose that G is a finite group and f is a complex-valued function on G. f induces a (left) convolution operator from L ²(G) to L ²(G) by g ? f *g{g \mapsto f \ast g} where

Region of Variability for Exponentially Convex Univalent Functions

For ${\alpha\in\mathbb C{\setminus}\{0\}}For a ? \mathbb C\{0}{\alpha\in\mathbb C{\setminus}\{0\}} let E(a){\mathcal{E}(\alpha)} denote the class of all univalent functions f in the unit disk \mathbbD{\mathbb{D}} and is given by f(z)=z+a₂z²+a₃z³+?{f(z)=z+a_2z^2+a_3z^3+\cdots}, satisfying

${\rm Re}\left (1+ \frac{zf'(z)}{f'(z)}+\alpha zf'(z)\right ) > 0 \quad {\rm in }\,{\mathbb D}.${\rm Re}\left (1+ \frac{zf'(z)}{f'(z)}+\alpha zf'(z)\right ) > 0 \quad {\rm in }\,{\mathbb D}.      5. Modified Roper-Suffridge operator for some holomorphic mappings      Shuxia FENG  Lin YU 《Frontiers of Mathematics in China》2011,6(3):411-426 In this paper, we will study the operator given by F(z) = (f(z1 ) + f￠(z1 )P(z0 ),(f￠(z1 ))1/k z0 T )T ,F(z) = (f(z_1 ) + f'(z_1 )P(z_0 ),(f'(z_1 ))^{1/k} z_0 ^T )^T ,      6. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]n{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cn{\mathbb {C}^n}. Consider the ring Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      7. Some Closed Range Integral Operators on Spaces of Analytic Functions      Austin Anderson 《Integral Equations and Operator Theory》2011,69(1):87-99 Our main result is a characterization of g for which the operator Sg(f)(z) = ò0z f￠(w)g(w) dw{S_g(f)(z) = \int_0^z f'(w)g(w)\, dw} is bounded below on the Bloch space. We point out analogous results for the Hardy space H 2 and the Bergman spaces A p for 1 ≤ p < ∞. We also show the companion operator Tg(f)(z) = ò0z f(w)g￠(w)  dw{T_g(f)(z) = \int_0^z f(w)g'(w) \, dw} is never bounded below on H 2, Bloch, nor BMOA, but may be bounded below on A p .      8. Smooth Universal Taylor Series      Ch. Kariofillis  Ch. Konstadilaki  V. Nestoridis 《Monatshefte für Mathematik》2006,87(5):249-257 Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A∞ (Ω). For g ∈ A∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote SN (g,z)(z) = ?Nl=0\fracg(l) (z)l ! (z-z)lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A∞(Ω), such that the following hold: i)  There exists a strictly increasing sequence μn ∈ {0, 1, 2, …}, n = 1, 2, …, such that, for every pair of compact sets Γ, Δ ⊂ [`(W)]\bar{\Omega} and every l ∈ {0, 1, 2, …} we have supz ? G supw ? D \frac?l?wl Smn ( f,z) (w)-f(l)(w) ? 0,    as n ? + ￥    and\sup_{\zeta \in \Gamma} \sup_{w \in \Delta} \frac{\partial^l}{\partial w^l} S_{\mu_ n} (\,f,\zeta) (w)-f^{(l)}(w) \rightarrow 0, \hskip 7.8pt {\rm as}\,n \rightarrow + \infty \quad {\rm and} ii)  For every compact set K ì \Bbb CK \subset {\Bbb C} with K?[`(W)] = ?K\cap \bar{\Omega} =\emptyset and Kc connected and every function h: K? \Bbb Ch: K\rightarrow {\Bbb C} continuous on K and holomorphic in K0, there exists a subsequence { m￠n }￥n=1\{ \mu^\prime _n \}^{\infty}_{n=1} of {mn }￥n=1\{\mu_n \}^{\infty}_{n=1} , such that, for every compact set L ì [`(W)]L \subset \bar{\Omega} we have supz ? L supz ? K Sm￠n ( f,z)(z)-h(z) ? 0,    as  n? + ￥.\sup_{\zeta \in L} \sup_{z\in K} S_{\mu^\prime _n} (\,f,\zeta )(z)-h(z) \rightarrow 0, \hskip 7.8pt {\rm as} \, n\rightarrow + \infty .       9. The composition operators on weighted Bloch space   总被引：9，自引：0，他引：9   R. Yoneda 《Archiv der Mathematik》2002,78(4):310-317 We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B log = { f ? H(D): supz ? D (1-| z|2) ( log\frac21-| z|2 )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.      10. Punishing Factors for Finitely Connected Domains      F. G. Avkhadiev  K.-J. Wirths 《Monatshefte für Mathematik》2006,106(1):103-115 Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities Cn(W,P) := supf ? A(W,P)supz ? W\frac|f(n)(z)| R(f(z),P)n! (R(z,W))nC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.      11. Isometric Equivalence of Integration Operators      Nadia J. Gal  James E. Jamison  Aristomenis G. Siskakis 《Complex Analysis and Operator Theory》2010,4(2):245-255 We are interested in the isometric equivalence problem for the Cesàro operator C(f) (z) = \frac1z ò0zf(x) \frac11-xd x{C(f) (z) =\frac{1}{z} \int_{0}^{z}f(\xi) \frac{1}{1-\xi}d \xi} and an operator Tg(f)(z)=\frac1zò0zf(x) g￠(x) d x{T_{g}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\xi) g^{\prime}(\xi) d \xi}, where g is an analytic function on the disc, on the Hardy and Bergman spaces. Then we generalize this to the isometric equivalence problem of two operators Tg1{T_{g_{1}}} and Tg2{T_{g_{2}}} on the Hardy space and Bergman space. We show that the operators Tg1{T_{g_{1}}} and Tg2{T_{g_{2}}} satisfy Tg1U1=U2Tg2{T_{g_{1}}U_{1}=U_{2}T_{g_{2}}} on H p , 1 ≤ p < ∞, p ≠ 2 if and only if g2(z) = lg1(eiqz){g_{2}(z) =\lambda g_{1}(e^{i\theta}z) }, where λ is a modulus one constant and U i , i = 1, 2 are surjective isometries of the Hardy Space. This is analogous to the Campbell-Wright result on isometrically equivalence of composition operators on the Hardy space.      12. A Universality Property of Gaussian Analytic Functions      Andrew Ledoan  Marco Merkli  Shannon Starr 《Journal of Theoretical Probability》2012,25(2):496-504 We consider random analytic functions defined on the unit disk of the complex plane f(z) = ?n=0￥ an Xn znf(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}, where the X n ’s are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients a n are chosen so that f(z) is defined on a domain of ℂ carrying a planar or hyperbolic geometry, and Ef(z)[`(f(w))]\mathbf{E}f(z)\overline{f(w)} is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virág. We show that for non-Gaussian coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically to the boundary of the domain. The proof is elementary and general.      13. Univalent Functions, <Emphasis Type="Italic">VMOA</Emphasis> and Related Spaces      Petros Galanopoulos  Daniel Girela  Rodrigo Hernández 《Journal of Geometric Analysis》2011,21(3):665-682 This paper is concerned mainly with the logarithmic Bloch space ℬlog  which consists of those functions f which are analytic in the unit disc \mathbbD{\mathbb{D}} and satisfy sup|z| < 1(1-|z|)log\frac11-|z||f￠(z)| < ￥\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty , and the analytic Besov spaces B p , 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of: •  A bounded univalent function in $\bigcup_{p>1}B^{p}$\bigcup_{p>1}B^{p} but not in the logarithmic Bloch space.      14. On properties of entire solutions of linear differential equations with polynomial coefficients      Ya. S. Mahola  M. M. Sheremeta 《Journal of Mathematical Sciences》2012,181(3):366-382 We investigate properties of entire solutions of differential equations of the form znw(n) + ?j = n - m + 1n - 1 an - j + 1(j)zjw(j) + ?j = 0n - m ( an - j - m + 1(j)zm + an - j + 1(j) )zjw(j) = 0, {z^n}{w^{(n)}} + \sum\limits_{j = n - m + 1}^{n - 1} {a_{n - j + 1}^{(j)}{z^j}{w^{(j)}}} + \sum\limits_{j = 0}^{n - m} {\left( {a_{n - j - m + 1}^{(j)}{z^m} + a_{n - j + 1}^{(j)}} \right){z^j}{w^{(j)}}} = 0,      15. On global transformations of ordinary differential equations of the second order      Václav Tryhuk 《Czechoslovak Mathematical Journal》2000,50(3):499-508 The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form f( t,uy,wy + uuz ) = f( x,y,z )u2 u+ g( t,x,u,u,w )uz + h( t,x,u,u,w )y + 2uwzf\left( {t,\upsilon y,wy + u\upsilon z} \right) = f\left( {x,y,z} \right)u^2 \upsilon + g\left( {t,x,u,\upsilon ,w} \right)\upsilon z + h\left( {t,x,u,\upsilon ,w} \right)y + 2uwz      16. A linear functional equation with a singularity at a fixed point      V. Mityushev 《Aequationes Mathematicae》1999,57(1):37-44 Summary. Local solutions of the functional equation¶¶zk f( z) = ?k=1nGk( z) f( skz ) +g( z) z{^\kappa} \phi \left( z\right) =\sum_{k=1}^nG_k\left( z\right) \phi \left( s_kz \right) +g\left( z\right) ¶with k > 0 \kappa > 0 and | sk| \gt 1 \left| s_k\right| \gt 1 are considered. We prove that the equation is solvable if and only if a certain system of k \kappa conditions on Gk (k = 1, 2, ... , n) and g is fulfilled.      17. Sharp A 2 inequality for Haar shift operators      Michael T. Lacey  Stefanie Petermichl  Maria Carmen Reguera 《Mathematische Annalen》2010,348(1):127-141 As a Corollary to the main result of the paper, we give a new proof of the inequality ||T f||L 2 (w) <~||w||A2||f||L 2(w) ,||{\rm T} f||_{L ^{2} (w)} \lesssim ||w{||_ {A_2}}||f{||_{{L ^2}(w)}}\,,      18. Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I      Stevan Pilipovi?  Nenad Teofanov  Joachim Toft 《Journal of Fourier Analysis and Applications》2011,17(3):374-407 Let ω,ω 0 be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WFFLq(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FLq(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S(w0)r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that $[b]{lll}\mathrm{WF}_{\mathcal{F}L^q_{(\omega /\omega _0)}}(\operatorname {Op}(a)f)&\subseteq&\mathrm{WF}_{\mathcal{F}L^q_{(\omega )}}(f)\\[6pt]&\subseteq&\mathrm{WF}_{\mathcal{F}L^q_{(\omega/\omega _0)}}(\operatorname {Op}(a)f)\cup \operatorname {Char}(a).$\begin{array}[b]{lll}\mathrm{WF}_{\mathcal{F}L^q_{(\omega /\omega _0)}}(\operatorname {Op}(a)f)&\subseteq&\mathrm{WF}_{\mathcal{F}L^q_{(\omega )}}(f)\\[6pt]&\subseteq&\mathrm{WF}_{\mathcal{F}L^q_{(\omega/\omega _0)}}(\operatorname {Op}(a)f)\cup \operatorname {Char}(a).\end{array}      19. Solving a class of variational inequalities with inexact oracle operators      Deren Han  Wei Xu  Hai Yang 《Mathematical Methods of Operations Research》2010,71(3):427-452 Consider a class of variational inequality problems of finding ${x^*\in S}Consider a class of variational inequality problems of finding x* ? S{x^*\in S}, such that f(x*)T (z-x*) 3 0,    "z ? S,f(x^*)^\top (z-x^*)\geq 0,\quad \forall z\in S,      20. An inverse spectral theorem for Kre?n strings with a negative eigenvalue      Harald Woracek 《Monatshefte für Mathematik》2012,33(3):105-149 A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L2(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D\mathfrakmDx{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as f￠(x) + z ò0L f(y) d\mathfrakm(y) = 0,    x ? \mathbb R, f￠(0-) = 0.f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Modified Roper-Suffridge operator for some holomorphic mappings

In this paper, we will study the operator given by

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Some Closed Range Integral Operators on Spaces of Analytic Functions

Our main result is a characterization of g for which the operator S_g(f)(z) = ò₀^z f￠(w)g(w) dw{S_g(f)(z) = \int_0^z f'(w)g(w)\, dw} is bounded below on the Bloch space. We point out analogous results for the Hardy space H ² and the Bergman spaces A ^p for 1 ≤ p < ∞. We also show the companion operator T_g(f)(z) = ò₀^z f(w)g￠(w)  dw{T_g(f)(z) = \int_0^z f(w)g'(w) \, dw} is never bounded below on H ², Bloch, nor BMOA, but may be bounded below on A ^p .     

Smooth Universal Taylor Series

Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote S_N (g,z)(z) = ?^N_l=0\fracg^(l) (z)l ! (z-z)^lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

The composition operators on weighted Bloch space

We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B _log = { f ? H(D): sup_{z ? D} (1-| z|²) ( log\frac21-| z|² )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

Isometric Equivalence of Integration Operators

We are interested in the isometric equivalence problem for the Cesàro operator C(f) (z) = \frac1z ò₀^zf(x) \frac11-xd x{C(f) (z) =\frac{1}{z} \int_{0}^{z}f(\xi) \frac{1}{1-\xi}d \xi} and an operator T_g(f)(z)=\frac1zò₀^zf(x) g^￠(x) d x{T_{g}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\xi) g^{\prime}(\xi) d \xi}, where g is an analytic function on the disc, on the Hardy and Bergman spaces. Then we generalize this to the isometric equivalence problem of two operators T_g1{T_{g_{1}}} and T_g2{T_{g_{2}}} on the Hardy space and Bergman space. We show that the operators T_g1{T_{g_{1}}} and T_g2{T_{g_{2}}} satisfy T_g1U₁=U₂T_g2{T_{g_{1}}U_{1}=U_{2}T_{g_{2}}} on H ^p , 1 ≤ p < ∞, p ≠ 2 if and only if g₂(z) = lg₁(e^iqz){g_{2}(z) =\lambda g_{1}(e^{i\theta}z) }, where λ is a modulus one constant and U _i , i = 1, 2 are surjective isometries of the Hardy Space. This is analogous to the Campbell-Wright result on isometrically equivalence of composition operators on the Hardy space.     

A Universality Property of Gaussian Analytic Functions

We consider random analytic functions defined on the unit disk of the complex plane f(z) = ?_n=0^￥ a_n X_n zⁿf(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}, where the X _n ’s are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients a _n are chosen so that f(z) is defined on a domain of ℂ carrying a planar or hyperbolic geometry, and Ef(z)[`(f(w))]\mathbf{E}f(z)\overline{f(w)} is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virág. We show that for non-Gaussian coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically to the boundary of the domain. The proof is elementary and general.     

Univalent Functions, <Emphasis Type="Italic">VMOA</Emphasis> and Related Spaces

This paper is concerned mainly with the logarithmic Bloch space ℬ_log which consists of those functions f which are analytic in the unit disc \mathbbD{\mathbb{D}} and satisfy sup_{|z| < 1}(1-|z|)log\frac11-|z||f^￠(z)| < ￥\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty , and the analytic Besov spaces B ^p , 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of:

On properties of entire solutions of linear differential equations with polynomial coefficients

We investigate properties of entire solutions of differential equations of the form

On global transformations of ordinary differential equations of the second order

The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form

A linear functional equation with a singularity at a fixed point

Summary. Local solutions of the functional equation¶¶z^k f( z) = ?_k=1ⁿG_k( z) f( s_kz ) +g( z) z{^\kappa} \phi \left( z\right) =\sum_{k=1}^nG_k\left( z\right) \phi \left( s_kz \right) +g\left( z\right) ¶with k > 0 \kappa > 0 and | s_k| \gt 1 \left| s_k\right| \gt 1 are considered. We prove that the equation is solvable if and only if a certain system of k \kappa conditions on G_k (k = 1, 2, ... , n) and g is fulfilled.     

Sharp A 2 inequality for Haar shift operators

As a Corollary to the main result of the paper, we give a new proof of the inequality

Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WF_FL^q(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FL^q_(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S^(w₀)_r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that

Solving a class of variational inequalities with inexact oracle operators

Consider a class of variational inequality problems of finding ${x^*\in S}Consider a class of variational inequality problems of finding x^* ? S{x^*\in S}, such that

f(x*)T (z-x*) 3 0,    "z ? S,f(x^*)^\top (z-x^*)\geq 0,\quad \forall z\in S,      20. An inverse spectral theorem for Kre?n strings with a negative eigenvalue      Harald Woracek 《Monatshefte für Mathematik》2012,33(3):105-149 A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L2(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D\mathfrakmDx{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as f￠(x) + z ò0L f(y) d\mathfrakm(y) = 0,    x ? \mathbb R, f￠(0-) = 0.f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.

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An inverse spectral theorem for Kre?n strings with a negative eigenvalue

A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L²(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D_\mathfrakmD_x{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as