Classification theorems for operators preserving zeros in a strip

We characterize all linear operators which preserve certain spaces of entire functions whose zeros lie in a closed strip. Necessary and sufficient conditions are obtained for the related problem with real entire functions, and some classical theorems of de Bruijn and Pólya are extended. Specifically, we reveal new differential operators which map real entire functions whose zeros lie in a strip to real entire functions whose zeros lie in a narrower strip; this is one of the properties that characterize a “strong universal factor” as defined by de Bruijn. Using elementary methods, we prove a theorem of de Bruijn and extend a theorem of de Bruijn and Ilieff which states a sufficient condition for a function to have a Fourier transform with only real zeros.     

On zeros of Hecke <Emphasis Type="Italic">L</Emphasis>-functions and their linear combinations on the critical line

In this paper two theorems were obtained. In the first theorem it is proved that a positive proportion of non-trivial zeros lie on the critical line for L-functions attached to automorphic cusp forms for congruence-subgroups. Therefore, the class of functions satisfying a variant of Selberg’s theorem was extended. In the second theorem a new lower bound was obtained for the number of zeros of linear combinations of Hecke L-functions on the intervals of the critical line. This theorem essentially improves the previously known S.A. Gritsenko’s result of 1997.     

Growth and distortion theorems on subclasses of quasi-convex mappings in several complex variables

In this paper, the refining growth and covering theorems for f are established, where f is a quasi-convex mapping of order α and x = 0 is a zero of order k + 1 of f(x) − x. As an application, we obtain the upper and lower bounds on the distortion theorem of f(x) defined on the unit polydisc of ℂ ⁿ . The upper bound of the distortion theorem for f(x) defined on the unit ball of a complex Banach space is also given. Our results extend the growth and distortion theorems for convex functions of one complex variable to quasi-convex mappings of several complex variables.     

Estimates of the number of zeros of some functions with algebraic Taylor coefficients

We prove two theorems about the number of zeros of analytic functions from certain classes that include the SiegelE-andG-functions. By using these theorems, we arrive at a new proof of the Gel'fond-Schneider theorem and improve the result that the numerical determinant does not vanish in the proof of the Shidlovskii theorem. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 817–824, June, 1997. Translated by M. A. Shishkova     

Convergence of Newton–Kantorovich Approximations to an Approximate Zero

In this paper, we prove an a posteriori and an a priori convergence theorem for Newton–Kantorovich approximations starting from an initial point x ₀. We apply these results to operators that are analytic at interior points of a closed ball centered at x ₀ and of radius R. We obtain some theorems on approximate zeros and on approximate zeros of second kind for these operators, which improve previous results.     

Instability of closed invariant sets of semidynamical systems: Method of sign-constant Lyapunov functions

In the paper, a reduction principle for the instability property of a closed positively invariant set M for semidynamical systems is proved. The fact that the result is not traditional is stressed by the assumption on the existence of a closed positively invariant set with respect to which the set M has the attraction property. The corresponding instability theorem of the method of sign-constant Lyapunov functions is presented. The assertion thus obtained generalizes the well-known Chetaev and Krasovskii theorems for systems of ordinary differential equations, theorems on the instability with respect to some of the variables, and also the Shimanov and Hale theorems for systems with retarded argument. Illustrating examples are presented.     

Erdős-Turʼan-Type Discrepancy for the Zeros of Rational Functions

In 1950 P. Erdős and P. Turán published a discrepancy theorem for the zeros of a polynomial. Therein, the maximum deviation of the normalized zero counting measure from the equilibrium measure of the unit circle is estimated. Many other discrepancy theorems and related propositions about weak-star-convergence of the zero distribution of a sequence of polynomials were proved during the last decades. For several years the weak-star-convergence of the zero distribution of a sequence of rational functions is also studied. The main result of this paper is a discrepancy theorem for the zero distribution of a rational function which generalizes and sharpens previous propositions about weak-star-convergence of the zero counting measure of sequences of rational functions and known discrepancy theorems for polynomials.     

Estimates for the orders of zeros of polynomials in some analytic functions

In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special D-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points.     

Convergence of Row Sequences of Simultaneous Padé–Faber Approximants

For meromorphic circumferentially mean p-valent functions, an analog of the classical distortion theorem is proved. It is shown that the existence of connected lemniscates of the function and a constraint on a cover of two given points lead to an inequality involving the Green energy of a discrete signedmeasure concentrated at the zeros of the given function and the absolute values of its derivatives at these zeros. This inequality is an equality for the superposition of a certain univalent function and an appropriate Zolotarev fraction.     

Bloch Constant of Holomorphic Mappings on the Unit Ball of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>*

In this paper, the authors establish distortion theorems for various subfamilies Hk(B) of holomorphic mappings defined in the unit ball in Cn with critical points, where k is any positive integer. In particular, the distortion theorem for locally biholomorphic mappings is obtained when k tends to oo. These distortion theorems give lower bound son det f(z) and Redet f'(z). As an application of these distortion theorems, the authors give lower and upper bounds of Bloch constants for the subfamilies βk (M) of holomorphic mappings. Moreover, these distortion theorems are sharp. When B is the unit disk in C, these theorems reduce to the results of Liu and Minda. A new distortion result of Re det f'(z) for locally biholomorphic mappings is also obtained.     

System of Generalized Vector Quasi-Equilibrium Problems in Locally FC-Spaces

A new class of locally finite continuous topological spaces （for short, locally FC-spaces） and a class of system of generalized vector quasi-equilibrium problems are introduced. By applying a generalized Himmelberg type fixed point theorem for a set-valued mapping with KKM-property due to the author, a collectively fixed point and an equilibrium existence theorem of generalized game are first proved in locally FC-spaces. By applying our equilibrium existence theorem of generalized game, some new existence theorems of equilibrium points for the system of generalized vector quasi-equilibrium problems are proved in locally FC-spaces. These theorems improve, unify and generalize many known results in the literatures.     

Generalized Knaster–Kuratowski–Mazurkiewicz Theorem Without Convex Hull

In this paper, a new notion of Knaster–Kuratowski–Mazurkiewicz mapping is introduced and a generalized Knaster–Kuratowski–Mazurkiewicz theorem is proved. As applications, some existence theorems of solutions for (vector) Ky Fan minimax inequality, Ky Fan section theorem, variational relation problems, n-person noncooperative game, and n-person noncooperative multiobjective game are obtained.     

A new look at some classical theorems on continuous functions on normal spaces

A sufficient condition for the strict insertion of a continuous function between two comparable upper and lower semicontinuous functions on a normal space is given. Among immediate corollaries are the classical insertion theorems of Michael and Dowker. Our insertion lemma also provides purely topological proofs of some standard results on closed subsets of normal spaces which normally depend upon uniform convergence of series of continuous functions. We also establish a Tietze-type extension theorem characterizing closed G _δ -sets in a normal space. This research was supported by the Ministry of Education and Science of Spain and FEDER under grant MTM2006-14925-C02-02. The first named author also acknowledges financial support from the University of the Basque Country under grant UPV05/101.     

CLT for linear random fields with stationary martingale-difference innovation

In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations were proved. In this paper, we present the central limit theorem for linear random fields with martingale-differences innovations satisfying the central limit theorem from [J. Dedecker, A central limit theorem for stationary random fields, Probab. Theory Relat. Fields, 110(3):397–426, 1998] and arranged in lexicographical order.     

Picard values of <Emphasis Type="Italic">p</Emphasis>-adic meromorphic functions

We investigate Picard-Hayman behavior of derivatives of meromorphic functions on an algebraically closed field K, complete with respect to a non-trivial ultrametric absolute value. We present an analogue of the well-known Hayman’s alternative theorem both in K and in any open disk. Here the main hypothesis is based on the behaviour of |f|(r) when r tends to +∞ on properties of special values and quasi-exceptional values.We apply this study to give some sufficient conditions on meromorphic functions so that they satisfy Hayman’s conjectures for n = 1and for n = 2. Given a meromorphic transcendental function f, at least one of the two functions f′f and f′f ² assumes all non-zero values infinitely often. Further, we establish that if the sequence of residues at simple poles of a meromorphic transcendental function on K admits no infinite stationary subsequence, then either f′ + af ² has infinitely many zeros that are not zeros of f for every a ∈ K* or both f′ + bf ³ and f′ + bf ⁴ have infinitely many zeros that are not zeros of f for all b ∈ K*. Most of results have a similar version for unbounded meromorphic functions inside an open disk.     

关于分担值的正规族和唯一性定理

设F是单位圆盘△上的亚纯函数族,α是一个非零的有穷复数,k是正整数,如果(?)f∈F,满足 1)f的零点重级≥k 1; 2)f和f(k)IM分担α,则F在△上正规. 此外,还证明了相应于正规函数以及整函数的唯一性定理方面的的结果.     

Transfer theorems and the algebra of modal operators

A set theory ZFI′ which does not employ the Law of the Excluded Middle φ ∀ ⊥ φ, for all φ, retians the stock of expressive capacities of the classical set theory ZF, on the one hand, and has many of the features of an effective theory on the other. In the article, a broad class of formulas σ is constructed for which ZF ⊥ σ implies ZFI′ ⊥ σ. This result provides a generalization of Friedman's theorem on AE-arithmetic formulas. Besides, we prove transfer theorems of classical logic for the case of rings; in particular, Hilbert's theorem on zeros and Artin's theorem on ordered fields are extended to the case of regular f-rings, and we bring in appropriate upper bounds for them. Supported by RFFR grant No. 93-012-1027. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 282–303, May–June, 1997.     

Accuracy of approximation of subharmonic functions by logarithms of moduli of analytic functions in the Chebyshev metrics

It is known that a subharmonic function of finite order ρ can be approximated by the logarithm of the modulus of an entire function at a point z outside an exceptional set up to C log |z|. In this paper, we prove that if such an approximation becomes more precise, i.e., the constant C decreases, then, beginning with C = ρ/4, the size of the exceptional set enlarges substantially. Similar results are proved for subharmonic functions of infinite order and for functions that are subharmonic in the unit disk. These theorems improve and complement a result by Yulmukhametov. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 55–73.     

Some topological minimax theorems

Two topological variants of the minimax theorem are proved with no restrictions on one of the spaces except for those related to the function under consideration. The conditions concerning the behavior of the function deal only with the interval between the maximin and minimax. As corollaries, we obtain the well-known theorems of Sion and Hoang-Tui on quasiconvex-quasiconcave semicontinuous functions. The scheme of arguments goes back to the Hahn-Banach theorem and the separating hyperplane theorem. It is shown how this scheme can be explicitly realized in the proof of the Hahn-Banach theorem. Translated fromMaternaticheskie Zametki, Vol. 67, No. 1, pp. 141–149, January, 2000.     

Extremal Problems of Function Theory Connected with the n-Fold Symmetry

We consider some conventional problems of the theory of functions of a complex variable such that their extremal configurations have the n-fold symmetry. We discuss two-point distortion theorems corresponding to the two-fold symmetry. New estimates are obtained for the module of a doubly connected domain. These estimates generalize known results by Rengel, Grötzsch, and Teichmüller to the case of rings with the n-fold symmetry, where . New distortion theorems are proved for functions meromorphic and univalent in a disk or in a ring. In these theorems, the extremal function also has the corresponding symmetry. All of the problems mentioned above are unified by the method applied; this method is based on properties of the conformal capacity and on symmetrization. Bibliography: 27 titles.