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.     

Operator-valued typically real functions

Generalizing the classical typically real functions in complex analysis, we introduce the operator-valued typically real functions and show how to construct these functions.

     

On a region of values in the class of typically real functions

The paper studies the regions of values of the systems {f(z₁), f(r₁), f(r₂),…, f(r_n)} and {f(r₁), f(r₂),…, f (r_n)}, where n ⁥ 2; z₁ is an arbitrary fixed point of the disk U = {z: |z| < 1} with Im z₁ ≠ 0; r_j are fixed numbers, 0 < r_j < 1, j = 1, 2,…, n; f ∈ T, and the class T consists of the functions f(z), f(0) = 0, f′(0) = 1, regular in the disk U and satisfying the condition Im f(z) · Imz > 0 for Im z ≠ 0. As an implication, the region of values of f(z₁) in the subclass of functions f ∈ T with prescribed values f(r_j) (j = 1, 2,…, n) is determined. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 5–16.     

A division theorem for real analytic functions

We characterize those homogeneous polynomials P [z₁, ... ,z_d] for which the principal ideal (P) = P · A(^d) is complementedin A(^d) or, equivalently, those which admit a continuous lineardivision operator. The condition is the same as that which characterizes,among the homogeneous polynomials, those which are nonellipticand for which P(D) is surjective in A(^d), and those for whichP(D) admits a continuous linear right inverse in C(^d). It dependsonly on the type of real singularities.     

On typically real functions which are generated by a fixed typically real function

Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ:= {z ∈ ℂ: |z| < 1}, normalized by f(0) = f′(0) − 1 = 0 and such that Imz Im f(z) ⩾ 0 for z ∈ Δ.     

The region of values of a system of functionals in the class of typically real functions

Let T_R be the class of functions that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z₀), f(r)} for z₀ ∈ ℝ, r∈(-1,1) is studied. The sets D_r={f(z₀):f∈T_R, f(r)=a} for −1≤r≤1 and Δ_r={(c₂, c₃): f ∈ T_R, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)⁻², r(1−r)⁻²) is an arbitrary fixed number. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79.     

A rotation theorem for the class of univalent p-symmetric functions

Exact estimates are obtained for the argument of the derivative on the class of all univalent holomorphic p-symmetric functions in the unit disk.Translated from Matematicheskie Zametki, Vol. 10, No. 2, pp. 239–242, August, 1971.     

A uniqueness theorem for functions in the extended Selberg class

Recently, we proved that every finite dimensional Alexandrov space is strongly locally Lipschitz contractible. In the present paper, we consider the set \(\mathcal M\) of all isometry classes of Alexandrov spaces of curvature \(\ge -1\) and of fixed dimension having upper diameter bound and lower volume bound, and prove that there exists a constant \(N\) depending on the parameters determining \(\mathcal M\) such that every space in \(\mathcal M\) can be covered by at most \(N\) strongly Lipschitz contractible balls. Also, we prove that there exists a constant \(N^\prime \) depending on \(\mathcal M\) such that every space in \(\mathcal M\) can be covered by at most \(N^\prime \) strongly Lipschitz contractible and convex regions.     

The region of values of f(z0) in a class of typically real functions

Let T_R be the class of functions f(z) with f(0)=0 and f(0)=1 that are regular and typically real in the disk ¦z¦< 1. The region of values of the system ª(z₀),f(r),f(0)/2} (for fixed z₀ and r, 0<r<1, on the class T_r is determined. The region of values of f(z₀) on the class of functions from T_r with fixed f(r) and f(0) is found. Bibliography:Dedicated to the 90th anniversary of the birth of my father, G. M. GoluzinTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 46–55.     

A class of functions of a real variable

An investigation of measurable almost-everywhere finite functions ξ(t), -∞ $$\varphi _T^\xi (\tau _{(n)} , \lambda _{(n)} ) = \frac{1}{{2T}}\int_{ - T}^T {\exp i} \sum\nolimits_{k - 1}^n {\lambda _k \xi (t - \tau _k )dt} $$ tends to an asymptotic characteristic function? _∞ ^ξ (τ _(n), λ_(n)) when T → ∞. Here n is any positive integer and T_(n)=(τ₁; τ₂, ..., τ_n) is arbitrary. It is proved that the class of such functions ξ(t) is larger than the class of Besicovich almost-periodic functions.