On some criteria for the univalence of analytic functions

Criteria for the univalence of functions f(z) which are regular in the domain ¦z¦ < 1 are obtained in the form of restrictions on the modulus of the quotient f (z) [f '(z)]^–1. Analogous results are obtained also for functions which are analytic in the interior of the unit disk except for a simple pole at infinity.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 359–366, March, 1973.In conclusion the author wishes to thank L. A. Aksent'ev and F. G. Avkhadiev for discussion of the results and valuable remarks.     

Some general univalence criteria for a family of integral operators

The main object of the present paper is to discuss some extensions of univalence conditions for a family of integral operators. Several other closely-related results are also considered. Many known univalence conditions are shown to follow upon specializing the parameters involved in our main results.     

Oscillation criteria for first-order neutral differential equations

This paper discusses a class of first-order neutral differential equations with variable coefficients and variable deviations. A series of sufficient conditions are established for all solutions of the equations to be oscillatory, and some of the conditions are sharp.     

The number of non-isomorphic models of an unstable first-order theory

It is proved that ifT is an unstable (first-order) theory,λ>|T|+ℵ₀, thenT has exactly 2 ^λ non-isomorphic models of cardinalityλ. In fact we have stronger results: this is true for pseudo-elementary classes, and for almost everyλ≧|T|+ℵ₁. The preparation of this paper was sponsored in part by NSF Grant GP-22937. This work was supported in part by NSF Grant GP-22794.     

Oscillation criteria for first-order neutral nonlinear difference equations with variable coefficients

Consider the first-order neutral nonlinear difference equation of the form

, where τ > 0, σ_i ≥ 0 (i = 1, 2,…, m) are integers, {p_n} and {q_n} are nonnegative sequences. We obtain new criteria for the oscillation of the above equation without the restrictions Σ_n=0^∞ q_n = ∞ or Σ_n=0^∞ nq_n Σ_j=n^∞ q_j = ∞ commonly used in the literature.     

Radius of univalence of holomorphic functions

Several simple observations are made regarding the radius of univalence of functions holomorphic in the disk ¦z¦<1 and also regarding the radius of holomorphicity of the inverse functions.Translated from Matematicheskie Zametki, Vol. 7, No. 3, pp. 295–298, March, 1970.     

On the univalence of an integral operator

In this work the author introduces a general integral operator and determines conditions for the univalence of this integral operator.     

On the univalence of the log-biharmonic mappings

In this paper, we establish the univalence and starlikeness connection between log-biharmonic mappings and logharmonic mappings.     

On univalence of two integral operators

In this work we consider two integral operators. The integral operators were constructed on the basis of the fact that the number of functions from the composition of the operators is the entire part of a complex number modulus.     

Complexity of deciding the first-order theory of real closed fields

Let denote an ordered field, where _i+1 is infinitesimal relative to the elements of the field _i 0 i < m (by definition, ₀=). Given a formula of the first-order theory of the real closed field _m, of the following form:, where P is a quantifier-free formula containing k atomic subformulas of the form (f_j 0), where are polynomials such that, and the absolute value of any integer occurring in the coefficients of the polynomials f_j is at most 2_M. Let =S₁+...+S_a denote the number of variables anda n the number of quantifiers in the formula. An algorithm is constructed which decides the truth of formulas of the above form in polynomial time with respect to M,. Previously known algorithms for this problem had complexity of the order of.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 53–100, 1988.