Regions of Values of the Systems {f(z 1), f(z 2), f'(z 2)} and {f(z 1), f'(z 1), f'(z 1)} on the Class of Typically Real Functions

Let T be the class of functions f(z) = z + a ₂ z ² + . . . that are regular in the unit disk and satisfy the condition Im f(z) Im z > 0 for Im z 0, and let z ₁ and z ₂ be any distinct fixed points in the disk |z| < 1. For the systems of functionals mentioned in the title, the regions of values on T are studied. As a corollary, the regions of values of f'(z ₂) and f'(z ₁) on the subclasses of functions in T with fixed values f (z ₁), f (z ₂) and f (z ₁), f'(z ₁), respectively, are found. Bibliography: 7 titles.     

The Value Regions of the Systems {f(z 1), f′(z 1)} and {f(z 1), f(z 2)} in the Class of Typically Real Functions

In the class T consisting of regular and typically real functions in the disk |x| < 1, the value regions of the system {f(z ₁), f(z ₁)} and {f(z ₁), f(z ₂)} are found for fixed z ₁ and z ₂. As an application, the value regions of f(z ₁) and f(z ₂) are found for f T with fixed value f(z ₁). Bibliography: 11 titles.     

Estimates for the zeros of differences of meromorphic functions

Let f be a transcendental meromorphic function and g(z)=f(z+c1)+f(z+c2)-2f(z) and g2(z)=f(z+c1)·f(z+c2)-f2(z).The exponents of convergence of zeros of differences g(z),g2(z),g(z)/f(z),and g2(z)/f2(z) are estimated accurately.     

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.     

Approximation by cubic splines in the classes of continuously differentiable functions

The problem of approximating continuously differentiable periodic functionsf(x) by cubic interpolation splines s_n(f; x) with equidistant nodes is considered. Asymptotically exact estimates for f(x)-s_n(f; x)_C are obtained in the classes of functions W¹H.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 215–226, February, 1972.In conclusion, I am deeply grateful to N. P. Korneichuk for a number of valuable remarks and conjectures utilized while working on this paper.     

Functions of bounded boundary rotation

Classes of functionsU _k, which generalize starlike functions in the same manner that the classV _k of functions with boundary rotation bounded bykπ generalizes convex functions, are defined. The radius of univalence and starlikeness is determined. The behavior off _α(z) = ∫ ₀ ^z [f'(t)]^α dt is determined for various classes of functions. It is shown that the image of |z|<1 underV _kfunctions contains the disc of radius 1/k centered at the origin, andV _k functions are continuous in |z|≦1 with the exception of at most [k/2+1] points on |z|=1.     

Accurate estimates of deviations of spline approximations to classes of differentiable functions

We derive the approximation on [0, 1] of functionsf(x) by interpolating spline-functions s_r(f; x) of degree 2r+1 and defect r+1 (r=1, 2,...). Exact estimates for ¦f(x)–s_r(f; x) ¦ and f(x)–s_r(f; x)|_c on the class W^mH for m=1, r=1, 2, ..., and m=2, 3, r=1 for the case of convex (t),are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 483–494, May, 1971.     

On power deformations of univalent functions

For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) − 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f _c (z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.     

On certain classes of analytic functions defined by Noor integral operator

Integral operator, introduced by Noor, is defined by using convolution. Let f_n(z)=z/(1−z)ⁿ⁺¹, n∈N₀, and let f be analytic in the unit disc E. Then I_nf=f⁽⁻¹⁾_n★f, where f_n★f⁽⁻¹⁾_n=z/(1−z). Using this operator, certain classes of analytic functions, related with the classes of functions with bounded boundary rotation and bounded boundary radius rotation, are defined and studied in detail. Some basic properties, rate of growth of coefficients, and a radius problem are investigated. It is shown that these classes are closed under convolution with convex functions. Most of the results are best possible in some sense.     

Approximation von extremalflächenstücken (hyperbolischen typs) durch charakteristische räumliche vierecke

We consider solutions z of the Cauchy-problem for hyperbolic Euler-Lagrange equations derived from a general Lagrangian f(x, y, z; z_x, z_y) in two independent variables x, y. z is supposed to be an extremal of the corresponding variational problem. Visualizing z as a surface in R ³ we give a geometric interpretation of Lewy's well-known characteristic approximation scheme for the numerical solution of second order hyperbolic equations by approximating z via a polyhedral construction built up from subunits which consist of two characteristic triangles having one side in common but lying on different planes in R ³. Utilizing ideas from Cartan-geometry one can (in an appropriate sense) introduce the “mean curvature” of these subunits and it is seen that this curvature vanishes (up to terms of higher order). That is a highly plausible result for the polyhedral approximation of “minimal surfaces”.     

Roots of the equationf (z)=αf (α) for the class of typically-real functions

Let T_r be the class of functionsf (z)=z+c₂z²+..., regular in the disk ¦z¦ <1, real on the diameter-1f (z) · Im z>0 in the remainder of the disk ¦z¦ <1. Let z _f be the solution off (z)= f (a) on Tr, where is any fixed complex number 0, 1, is any fixed real number, ¦¦< 1. We determine the region of values of the functional z_f on the class T_r. Variation formulas for Stieltjes integrals due to G.M. Goluzin are used.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 41–52, July, 1971.     

A Quantitative Voronovskaya Formula for Mellin Convolution Operators

Here we give a quantitative Voronovskaya formula for a class of Mellin convolution operators of type

Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks

Given a Brownian motion (B _t) _t0 in R ^d and a measurable real function f on R ^d belonging to the Kato class, we show that 1/t ₀ ^t f(B _s ) ds converges to a constant z with an exponential rate in probability if and only if f has a uniform mean z. A similar result is also established in the case of random walks.     

Metric diophantine approximation in Julia sets of expanding rational maps

Let T : J → J be an expanding rational map of the Riemann sphere acting on its Julia set J andf : J →R denote a Hölder continuous function satisfyingf(x)?log | T′(x vb for allx in J. Then for any pointz ₀ in J define the set D_{z 0}(f) of “well-approximable” points to be the set of points in J which lie in the Euclidean ball $B(\gamma ,{\text{ exp(}} - \sum {_{i - 0}^{\mathfrak{n} - 1} } f(T^\ell x)))$ for infinitely many pairs (y, n) satisfying T ⁿ (y)=z₀. We prove that the Hausdorff dimension of D_{z 0}(f) is the unique positive numbers(f) satisfying the equation P(T,?s(f).f)=0, where P is the pressure on the Julia set. This result is then shown to have consequences for the limsups of ergodic averages of Hölder continuous functions. We also obtain local counting results which are analogous to the orbital counting results in the theory of Kleinian groups.     

Solutions with boundary‐layers and spike‐layers to singularly perturbed quasilinear Dirichlet problems

The structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form –?Δ_pu = f(u) in Ω, u = 0 on ?Ω, Ω ? R ^N a bounded smooth domain, is studied as ? → 0⁺, for a class of nonlinearities f(u) satisfying f(0) = f(z₁) = f(z₂) = 0 with 0 < z₁ < z₂, f < 0 in (0, z₁), f > 0 in (z₁, z₂) and f(u)/u^p–1 = –∞. It is shown that there are many nontrivial nonnegative solutions with spike‐layers. Moreover, the measure of each spike‐layer is estimated as ? → 0⁺. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0,∞). Uniqueness of a solution with a boundary‐layer and many positive intermediate solutions with spike‐layers are obtained for ? sufficiently small. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonfree actions of countable groups and their characters

The paper studies the region of values of the system {c ₂, c ₃, f(z ₁), f′(z ₁)},where z ₁ is an arbitrary fixed point of the disk |z| < 1; f ∈ T,and the class T consists of all the functions f(z) = z + c ₂ z ² + c ₃z³ + ⋯ regular in the disk |z| < 1 that satisfy the condition Im z · Im f(z) > 0 for Im z ≠ 0. The region of values of f′(z ₁) in the subclass of functions f ∈ T with prescribed values c ₂, c ₃, and f(z ₁) is determined. Bibliography: 10 titles.     

Rigidity of codimension two indefinite spheres

We describe all isometric immersions f:S _n ^s S _s ⁿ +2/S _s ⁿ n–s4, whenever the set of totally geodesic points does not disconnect S _s ⁿ , where S _n ^s denotes the complete n-dimensional indefinite Riemannian space form of constant positive curvature 1 and signature s.     

To distortion theorems for typically real functions

The paper studies the region of values of the system {f(z ₁), f(z ₂), c ₂},where z _j , j=1, 2, are arbitrary fixed points of the disk |z|<1; f ∈ T, and the class T consists of all functions f(z) = z + c ₂ z ² + ··· regular in the disk |z| < 1 and satisfying the condition Im f(z)·Im z>0 for Im z > 0 for Im z ≠ 0. The region of values of f(z ₁) in the subclass of functions f (z) ∈ T with prescribed values c ₂ and f(z ₂) is determined. Bibliography: 8 titles.     

A survey of truncation error analysis for Padé and continued fraction approximants

To compute the value of a functionf(z) in the complex domain by means of a converging sequence of rational approximants {f _n(z)} of a continued fraction and/or Padé table, it is essential to have sharp estimates of the truncation error ¦f(z)–f _n(z)¦. This paper is an expository survey of constructive methods for obtaining such truncation error bounds. For most cases dealt with, {f _n(z)} is the sequence of approximants of a continued fractoin, and eachf _n(z) is a (1-point or 2-point) Padé approximant. To provide a common framework that applies to rational approximantf _n(z) that may or may not be successive approximants of a continued fraction, we introduce linear fractional approximant sequences (LFASs). Truncation error bounds are included for a large number of classes of LFASs, most of which contain representations of important functions and constants used in mathematics, statistics, engineering and the physical sciences. An extensive bibliography is given at the end of the paper.Research supported in part by the U.S. National Science Foundation under Grants INT-9113400 and DMS-9302584.     

On the Uniform Approximation of a Class of Analytic Functions by Bruwier Series

For a class of analytic functions f(z) defined by Laplace–Stieltjes integrals the uniform convergence on compact subsets of the complex plane of the Bruwier series (B-series) ∑^∞_n=0 λ_n(f) , λ_n(f)=f⁽ⁿ⁾(nc)+cf⁽ⁿ⁺¹⁾(nc), generated by f(z) and the uniform approximation of the generating function f(z) by its B-series in cones |arg z|< is shown.