The value regions of initial coefficients in a certain class of meromorphic functions

Let M_k,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+a_o+a₁z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition where J_λ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class M_k,λ we consider sorne coefficient problems and problems concerning distortion theorems. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 91–96. Translated by N. Yu. Netsvetaev.     

Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥_p=(∫ _a ^b |w(t)|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.     

On extremal properties for the polar derivative of polynomials

If p(z) is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then it is proved[5] that max |z|=1 |p′(z)| ≤ kn1n + kn m|z|=ax1 |p(z)|. In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type p(z) = cnzn + ∑n j=μ cn jzn j, 1 ≤μ≤ n. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.     

Duals of holomorphic Besov spaces on the polydisks and diagonal mappings

Let U ⁿ be the unit polydisk in C ⁿ and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω ₁, ..., ω _n ), ω _j ∈ S(1 ≤ j ≤ n) and f ∈ H(U ⁿ ). The function f is said to be in holomorphic Besov space B _p (ω) if

De Bruijn’s question on the zeros of Fourier transforms

Letf(z) be a real entire function of genus 1^*, δ≥0, and suppose that for each ε>0, all but a finite number of the zeros off(z) lie in the strip |Imz| ≤δ+ε. Let λ be a positive constant such that . It is shown that for each ε>0, all but a finite number of the zeros of the entire function lie in the strip and if Δ² < 2λ, then all but a finite number of the zeros of e^−λD2 f(z) are real and simple. As a consequence, de Bruijn's question whether the functions e^γ t ²,λ>0, are strong universal factors is answered affirmatively. The authors wish to acknowledge the financial support of the Korea Research Foundation made in the program year of (1998–2000).     

L p inequalities for polynomials with restricted zeros

LetP(z) be a polynomial of degreen which does not vanish in the disk |z|<k. It has been proved that for eachp>0 andk≥1, $$\begin{gathered} \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P^{(s)} (e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} \leqslant n(n - 1) \cdots (n - s + 1) B_p \hfill \\ \times \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P(e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} , \hfill \\ \end{gathered} $$ where $B_p = \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {k^s + e^{i\alpha } } \right|^p d\alpha } } \right\}^{ - 1/p} $ andP ^(s)(z) is thesth derivative ofP(z). This result generalizes well-known inequality due to De Bruijn. Asp→∞, it gives an inequality due to Govil and Rahman which as a special case gives a result conjectured by Erdös and first proved by Lax.     

Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions

Let U(λ, μ) denote the class of all normalized analytic functions f in the unit disk |z| < 1 satisfying the condition

Typically real mean univalent functions of large growth

Typically real normalized κ-dimensionally mean univalent functionsf on |z|<1 are considered for which $\mathop {\lim \sup }\limits_{r \uparrow 1} \{ (1 - r)^2 \mathop {\max }\limits_{0 \leqslant \theta \leqslant 2\pi } \left| {f(re^{i\theta } )} \right|\} > 0$ . Lets=logf(z) forz in the unit disc cut along (?1, 0]. A theorem is proved concerning the area of the Riemann surface over thes-plane which distinguishes the two cases ?1≦κ<+∞ and κ=+∞.     

Some integral inequalities for the polar derivative of a polynomial

If P(z) is a polynomial of degree n which does not vanish in |z| 1,then it is recently proved by Rather [Jour.Ineq.Pure and Appl.Math.,9 (2008),Issue 4,Art.103] that for every γ 0 and every real or complex number α with |α|≥ 1,{∫02π |D α P(e iθ)| γ dθ}1/γ≤ n(|α| + 1)C γ{∫02π|P(eiθ)| γ dθ}1/γ,C γ ={1/2π∫0 2π|1+eiβ|γdβ}-1/γ,where D α P(z) denotes the polar derivative of P(z) with respect to α.In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J.Approx.Theory,54 (1988),306-313] as a special case.     

Composition operators on the Lipschitz space in polydiscs

Let Un be the unit polydisc of Cn and φ= (φ1,...,φn? a holomorphic self-map of Un. Let 0≤α< 1. This paper shows that the composition operator Cφ, is bounded on the Lipschitz space Lipa(Un) if and only if there exists M > 0 such thatfor z∈Un. Moreover Cφ is compact on Lipa(Un) if and only if Cφ is bounded on Lipa(Un) and for every ε > 0, there exists a δ > 0 such that whenever dist(φ(z),σUn) <δ     

Some remarks on the identity between a variational integral and its relaxed functional

Summary Letf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] be measurable inx and convex inz. It is proved, by an example, that even iff verifies a condition as|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) with 1<p<q,a∈L _loc ^s (R ⁿ),s>1, the functional that isL ¹(Ω)-lower semicontinuous onW ^1,1(Ω), does not agree onW ^1,1(Ω) with its relaxed functional in the topologyL ¹(Ω) given by inf

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Riassunto Siaf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) con 1<p<q,a∈L _loc ^s (R ⁿ),s>1, il funzionale , che èL ¹(Ω)-semicontinuo inferiormente suW ^1,1(Ω), non coincide suW ^1,1(Ω) con il suo funzionale rilassato nella topologiaL ¹(Ω) definito da inf

     

Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions

For a cubature formula of the form $$\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {f(x,y)dxdy = \frac{{4\pi ^2 }} {{mn}}\sum\limits_{i = 0}^{n - 1} {\sum\limits_{j = 0}^{m - 1} {f\left( {\frac{{2\pi i}} {n},\frac{{2\pi j}} {m}} \right) + R_{n,m} (f)} } } }$$ on a Chebyshev grid, the remainder R _n,m (f) is proved to satisfy the sharp estimate $$\mathop {\sup }\limits_{f \in H\left( {r_1 ,r_2 } \right)} \left| {R_{n,m} (f)} \right| = O\left( {n^{ - r_1 + 1} + m^{ - r_1 + 1} } \right)$$ in some class of functions H(r ₁, r ₂) defined by a generalized shift operator. Here, r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in the O-term depends only on ??.     

Boundary distortion and variation of the module under an extension of a doubly connected domain

Theorem 2

Let f(z) ∈ $\mathcal{F}(\rho ,r)$ , f(z) ≠ e^iα f(z;pr), α ∈ ?, and let ?(t) be a strictly convex monotone function of t>0. Then $$\int\limits_0^{2\pi } {\Phi (|f'(e^{i\theta } )|)d\theta< } \int\limits_0^{2\pi } {\Phi (|f'(e^{i\theta } ;\rho ,r)|)d\theta } $$ . The proof of this theorem is based on the Golusin-Komatu equation. If E is a continuum in the disk U_R={z:|z|<R}, then M (R, E) denotes the conformal module of the doubly connected component of U_R/E; let $\varepsilon (m) = \{ E:\overline U _r \subset E \subset U_1 , M(1,E) = M^{ - 1} \} $ .

Problem 3

Find the maximum of M(R, E), R>1, and the minimum of cap E over all E in ε(m). This problem was posed by V. V. Kozevnikov in a lecture to the Seminar on Geometric Function Theory at the Kuban University in 1980, and by D. Gaier (see [2]). The solution of this problem is given by the following theorem.

Theorem 3

Let $E^* = \underline U _m \cup [m,s]$ . If R>1; E, E^* ∈ ε(m) and E ≠ e^iα E*, α ∈ ?, then M(R, E)<M(R, E^*), capE^*<capE. A similar statement is also proved for continua lying in the half-plane. Bibliography: 7 titles.     

Asymptotic property of approximation to x αsgn x by Newman Type Operators

Approximation to the function ｜x｜ plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals ｜x｜ if α = 1. We construct a Newman Type Operator rn（x） and prove max ｜x｜≤1｜xαsgn x-rn（x）｜～Cn1/4e-π1/2（1/2）αn.     

Bulk universality holds in measure for compactly supported measures

Let μ be a measure with compact support, with orthonormal polynomials {p _n } and associated reproducing kernels {K _n }. We show that bulk universality holds in measure in {ξ: μ′(ξ) > 0}. More precisely, given ɛ, r > 0, the linear Lebesgue measure of the set {ξ: μ′(ξ) > 0} and for which

On the product of the distances of a point from the vertices of a polytope

Letx ₁,...,x _m be points in the solid unit sphere ofE _n and letx belong to the convex hull ofx ₁,...,x _m. Then . This implies that all such products are bounded by (2/m) ^m (m −1) ^m−1. Bounds are also given for other normed linear spaces. As an application a bound is obtained for |p(z ₀)| where andp′(z ₀)=0.     

On a complete description of the class of functions without zeros analytic in a disk and having given orders

For arbitrary 0 ≤ σ ≤ ρ ≤ σ + 1, we describe the class A _σ ^ρ of functions g(z) analytic in the unit disk = {z : ∣z∣ < 1} and such that g(z) ≠ 0, ρ_T[g] = σ, and ρ_M[g] = ρ, where

On approximation of periodic functions by Fourier sums

Let L _p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||_p = ( ò _{- p}^p | f |^p )^{1 \mathord}

Convergence rates of the LIL for random fields in Hilbert spaces

Considering the positive d-dimensional lattice point Z ₊ ^d (d ≥ 2) with partial ordering ≤, let {X _k: k ∈ Z ₊ ^d } be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $ S_n = \sum\limits_{k \leqslant n} {X_k } $ S_n = \sum\limits_{k \leqslant n} {X_k } , n ∈ Z ₊ ^d . Let σ _i ², i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ ². Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) . We show that when l ≥ 2 and b > −l/2, E[‖X‖²(log ‖X‖) ^d−2(log log ‖X‖) ^b+4] < ∞ implies $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} , where Γ(·) is the Gamma function and $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } .     

Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$

In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f ₁(z), f ₂(z), …, f _n (z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ ⁿ and