On the minimum modulus of a multiple Dirichlet series with monotone coefficients

We establish conditions under which the relation M(x, F) (x, F) m(x, F) holds except for a small set, as ¦x¦ + for an entire function F(z) of several complex variables z (p2) represented by a Dirichlet series, where M(x, F) = sup{¦F(x+iy¦: y ^p}, m(x, F) = inf{¦F(x+iy)¦: y ^p} (x, F) being the maximal term of the Dirichlet series, and x ^p.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 21–25.     

Some inequalities of the V. A. Markov type

Let B be a domain in the complex plane, let p_n(z) and P_n(z) be polynomials of degree n where the zeros of P_n(z) lie in , let(z) be a finite function,(z) 0, z . We consider the problem of estimating from above the functions L[p_n(z)]=(z)p_n(z) – wp_n(z), z , if ¦p_n(z)¦ ¦P_n(z)¦ for zB. Under some very general conditions on B, z, (z), and w we prove the inequality ¦L[p_n(z)]¦ ¦L[P_n(z)]¦.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 431–440, April, 1968.     

A certain class of functions that are univalent in an annulus

In the class F₁ of functions f(), regular and univalent in the annulus ={<||<1} and satisfying the conditions ¦f()¦ < 1 and f() 0 for , ¦f()¦=1 ¦¦=1, for f(l)=1, one finds the set of the values D(A)=f(A): f for an arbitrary fixed point A. One makes use of the method of variations and certain facts from the theory of the moduli of families of curves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 82–92, 1985.     

Defining a metric in a linear space by means of a family of subsets

Necessary and sufficient conditions are given on a familyA _r ^r>0 of subsets of a real linear space X under which infr > 0 x A _r is a quasinorm [l] on X. It is shown that for any symmetric (about zero) closed set A in a normed space X containing the ball {x X: x l there exists a quasinorm ¦·¦ on X such that A = {x X ¦x¦ 1}. Examples are constructed of linear metric spaces in which there exists a Chebyshev line which is not an approximately compact set.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 237–246, February, 1976.     

The question of A*-summability of double trigonometric Fourier series

It is proved that for anyf(x, y) L(R), where R=[-,,-, ], a function (x, y), exists such that ¦(x, y) ¦=¦f(x, y) ¦ for almost all (x, y) R. The Fourier series of the function (x, y) and all conjugate trigonometric series are A^*-summable almost everywhere.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 145–150, February, 1972.     

On the set of positive deflections of functions meromorphic in the unit disk

For an arbitrary (0, 1/2) we construct a functionf(z) of lower order [f]=. that is meromorphic in the disk D = {z ¦z¦< 1} and such that the set (f) of positive deflections of the functionf (in the sense of V. P. Petrenko) has positive logarithmic capacity.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 48–53.     

Solution of two sequencing problems

The solution of the following problems is offered. Suppose a multiset J (¦J¦=p) is given. For each pair of elements and J, a number 1 P is given. Moreover, if 1 < x<p then x is undefined. If x=1, then x=p. Problem 1. Find the permutation ₁..._F of elements of the multiset J satisfying the following conditions. Let _i, _i=. If i,j < x, thenj <i. If i,j > x, then i<j. Such a permutation is called a PC-schedule. Problem 2. Find a PC-schedule in which the following property holds: if i < x < j, _i=, _j=, then. Such a PC-schedule is called an SC-schedule. The conditions under which these problems have solutions are studied. For their solution an algorithm of shifts is used with the complexity O(¦B(J)¦²¦J¦).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 44–72, 1983.     

Diophantine complexity

It is well known that any recursively enumerable set S of natural numbers can be represented by formulas of the following types a S x yx, z₁...Z_n (D (a,x,y,z₁,...,z_n)=0)¦(Davis normal form); aS z₁...z_n (E (a,z₁..., z_n)=E₂(a,z₁,...,z_n)) (exponential diophantine representation); and aS z₁... z_n, (D (a-, z₁),...)Z_n)=0) (diophantine representation). Each of these three representations yields a different measure for the complexity of S as a whole and the complexity of accepting individual members of S. A survey is presented of various results concerning such complexity measures, due to different authors.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeliniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 122–131, 1988.     

Regularization of the boundary-value problem for Sturm-Liouville equations with operator-valued coefficients

A regularization method is used to find an approximate solution to the inhomogeneous boundary-value problem –y(t)+Ay(t)=y(t)+h(t), t [0, b], cos CY–sin CY= where A is a self- adjoint unbounded operator on a Hilbert space H; C is an operator on H H; H H; h(t) L ₂ (H, (0, b)); Y, Y are regularizing boundary values of y (t).Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 23–28, 1989.     

A relation between Chung's and Strassen's laws of the iterated logarithm

Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)^–1/2–f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)^–1/2 W(xT)¦(loglog T)^–1]=/4 a.s. T 0x1     

An addendum to the paper “ranges of values of systems of functionals in certain classes of regular functions”

One gives an application of the results of the paper mentioned in the title to the problem of the range of the system of initial coefficients on the class V_k,(k2, 0<1) of functions f(z)=z+a₂z²+..., regular in ¦z¦<1, f(z)f(z)/z0 in ¦z¦<1, satisfying the condition ₀ ² ReJ(eⁱ)dk, 0<<1, where J(z)=(1+zf(z)/f(z)) +(1–+)zf(z)/f(z).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 31–35, 1986.     

Localization of the spectrum of certain non-self-adjoint operators

Let the self-adjoint operator A and the bounded operator B be specified in Hilbert space We let denote the spectral family of the operator A. If (E – E _N ) B ₂+E–_NB ² 0 npnN , then in the complex plane z=+ there will exist the curve ¦ ¦ =f (), limf () = 0 for ± such that the entire spectrum of the operator A+B lies within the region ¦ ¦ f(). In particular, the condition of the theorem will be satisfied when B is a completely continuous operator.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 415–420, April, 1968.The author expresses his appreciation to R. S. Ismagilov for his discussion of the results.     

Extremal problems in the class σ(Τ)

Let L_f(r)={w=f(x), ¦z¦=r}, 1 < r < , be a level line of the function f(z) . Sharp upper bounds are obtained for the diameter of the curve L_f(r) in the class () for functions f(z)=z+₀+₁z^–1+... for which there exists a domain _f that complements the exterior of the unit disk and has conformal radius at the point w=0 satisfying the condition R(_f,0) , 0 < < 1. Also, a set of values is found for the coefficient ₁ in the class ().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 138–153, 1991.     

A study of the rate of rational approximation of a class of analytic functions

This paper is devoted to results belonging to the theory of rational approximation of analytic functions. We study the rate of decrease of the best approximations _n of a functionf holomorphic in a disk ¦z ¦z¦< , >1> by rational functions of order at most n in the uniform metric on the unit disk with center at z=0. We prove theorems that connect the rate of decrease of the quantities _n with the order 0 and type 0 of the functionf. The proofs of these results are based on the methods of the theory of Hankel operators.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 32–39.The author is grateful to Prof. V. A. Prokhorov for posing this problem and assistance with the preparation of this article.     

Lp-convergence for expansions in terms of the eigenfunctions of a Sturm — Liouville problem

For the operator L_v=–(x^2ay). x [0, 1], y(0)=y(1)=0 with 0 < 1/2, or ¦y¦ < , y(1)=0 with 1/2 <1, we investigate the effect which the singularity of the Sturm-Liouville operator derived from this self-adjoint expression has on L_p-convergence of expansions in terms of the eigenfunctions of this operator. We will prove that the orthonormalized system of eigenfunctions forms a basis in L_p [0, 1] for 2/(2–) < p < 2/.Translated from Matematicheskii Zametki, Vol. 3, No. 6, pp. 683–692, June, 1968.The author is grateful to V. M. Tikhomirov for his many valuable remarks and his constant attention to this work.     

Quasiclassical asymptotic behavior of the scattering cross section for asymptotically homogeneous potentials

One considers the total scattering cross section on the potential gV(x), x^m, m3, for large values of the coupling constant g and of the wave number k. One assumes that V(x)(x/|1x|)|x|^–, 2>m+1, as ¦x¦. It is shown that for gk^–1 , g^3–ak^2(a–2) the scattering cross section is equal asymptotically to _a(gk^–1), x=(m–1)(–1)^–1. Here the coefficient _a is determined only by the function and the number . Under the additional conditions >0, V>0, the indicated asymptotic behavior holds in the large domain gk^–1 , gk^a–z c(gk^–1), >0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 105–136, 1986.     

Coefficients of univalent functions which assume no pair of values W and −W

In this paper we study the behavior of the coefficients of functions univalent in the disk ¦z¦< 1 and assuming there are no pair of values Wand –W. In particular, we establish the asymptotic behavior of b_n (n); for the coefficients we obtain the estimate ¦b_n¦ < 2.34 exp {l/4n} (n = 2,3, ...) and for each function of the class indicated we prove, subject to a certain condition, the relationship b_n+1¦–¦b_n=O(n^–1/2).Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 3–14, January, 1972.     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

Ranges of certain system of functionals in classes of functions with a positive real part

We give the algebraic characteristics of the range of the system C_p, C_2p, ..., c_np f() ( fixed, 0<¦¦<1, n1, P=1, 2, ...) on certain subclasses C_m,p, of the class C of functions, regular in the circle ¦z¦<1 and satisfying in it the condition Re f(z)>0. As an application one finds the range of f() on the subclasses C _m,p, ⁽ⁿ⁾ of functions from C_m,p, with prescribed coefficients c_p c_2p, ..., c_np.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 17–25, 1980.     

Bonds for a functional on a class of functions with positive real parts

We establish sharp upper and lower bounds for the functionalJ (p)- Re{p (z)-zp'(z)/p(z), 1. where ¦z¦=r is fixed, 0 <r <1, p(z).P_n (A, B), –1 B < a 1, a certain class of regular functions in the disk with values in the right halfplane.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 686–689, May, 1990.