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.     

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.     

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.     

The region of values of the system { f ( z 1),..., f ( z n )} in the class of typically real functions. III

The paper studies the region of values D_m,n(T) of the system {f(z₁), f(z₂),..., f(z_m), f(r₁), f(r₂),..., f(r_n)}, where m ≥ 1; n > 1; z_j, j = 1, ... m, are arbitrary fixed points of the disk U = {z: |z| < 1} with Im z_j ≠ 0, j = 1, 2, ..., m; r_j, 0 < r_j < 1, j = 1, 2, ..., n, are fixed; f ∈ T, and the class T consists of functions f(z) = z + c₂z² + ... regular in the disk U and satisfying the condition Im f(z) · Im z > 0 for Im z ≠= 0, z ∈ U. An algebraic characterization of the set D_m,n(T) in terms of nonnegative-definite Hermitian forms is provided, and all the boundary functions are described. 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, 3) is determined. Bibliography: 12 titles. Dedicated to the 100th anniversary of my father’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 23–34.     

Commutator equations in free groups

Letf ₁, …,f _n be free generators of a free groupF. We consider the equation [z ₁, …,z _n]_ω. where ω and ω′ indicate the disposition of brackets in the higher commutators [z ₁, …,z _n]_ω and [f ₁, …,f _n]_ω. We give a necessary and sufficient condition on ω and ω′ for the existence of solutions of this equation. It is also shown that for any solutionz ₁=r₁, …,z _z=r _n we have <r ₁, …,r _n>=〈f ₁, …f _n〉.     

The region of values of the system {ƒ(z1), …, ƒ(zn)} in the class of typically real functions. II

The paper studies the region of values D_m,1(T) of the system {ƒ(z₁), ƒ(z₂), …, ƒ(zm), ƒ(r)}, m e 1, where z_j (j = 1, 2, …,m) are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0 (j = 1, 2, …,m), and r, 0 < r < 1, is fixed, in the class T of functions ƒ(z) = z+a₂z²+ ⋯ regular in the disk U and satisfying in the latter the condition Im ƒ(z) Imz > 0 for Im z ≠ 0. An algebraic characterization of the set D_m,1(T) in terms of nonnegative-definite Hermitian forms is given, and all the boundary functions are described. As an implication, the region of values of ƒ(z_m) in the subclass of functions from the class T with prescribed values ƒ(z_k) (k = 1, 2, …,m − 1) and ƒ(r) is determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 24–33. Original article submitted June 13, 2005.     

A unified approach for univalent functions with negative coefficients using the Hadamard product

For given analytic functions ϕ(z) = z + Σ _n=2^∞ λ _n z ⁿ , Ψ(z) = z + Σ _n=2^∞ μ with λ _n ≥ 0, μ _n ≥ 0, and λ _n ≥ μ _n and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ(z) = z + Σ _n=2^∞ a _n z ⁿ in U such that f(z)*ψ(z)≠0 and

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP _n (z, f, f′)=P _n−1(z, f, f′, ... f ^(p)), whereP _n andP _n−1 are polynomials in all variables, the degree ofP _n with respect tof andf′ is equal ton, and the degree ofP _n−1 with respect tof, f′, ... f ^(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E _*, whereE _* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z ^1/2 (β+o(1)), β∈C, holds forz=re ^iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re ^iν)‖=o(r ^1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.     

Cyclic elements of the shift operator in weighted anisotropic spaces of holomorphic functions in the polydisk

Let ϕ(r) = (ϕ₁(r₁), …, ϕ_n(r_n)) be a vector-valued function on R ₊ ⁿ . A necessary and sufficient condition is obtained under which any function f ∈, H^∞ (D ⁿ ), f(z) ≠ 0, z ∈, D ⁿ , is cyclic in the corresponding weighted space L^p(ϕ), where D ⁿ is the unit polydisk in C ⁿ. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 226–234.     

Wavelet expansions of fractal measures

Let μ be a measure on ℝⁿ that satisfies the estimate μ(B _r(x))≤cr ^α for allx ∈ ℝⁿ and allr ≤ 1 (B _r(x) denotes the ball of radius r centered atx. Let ϕ _j,k ^(ɛ) (x)=2 ^nj2ϕ^(ɛ)(2 ^j x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤⁿ, and ∈ ∈E, a finite set, and letP _j (T)=Σ_ɛ,k <T,ϕ _j,k ^(ɛ) >ϕ _j,k ^(ɛ) denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls ^p(dμ) for 1 ≤p ≤ ∞, we show thatP _j(f dμ) ∈ L^p(dx) and ||P _j (fdμ)||L _p(dx)≤c2 ^{j((n-α)/p′))}||f||L _p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P _j (fdμ)||L _p(dx) under more restrictive hypotheses. Communicated by Guido Weiss     

Asymptotic separation for independent trajectories of Markov processes

We say that n independent trajectories ξ₁(t),…,ξ _n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ _i (t _i ) and ξ _j (t _j ) is at least ɛ, for some indices i, j and for all large enough t ₁,…,t _n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct ^−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ^(α) are asymptotically separated, where ξ^(α) is the α-process generated by −(−Δ)^α/2. Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000 RID="*" ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782 RID="**" ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573     

The primes contain arbitrarily long polynomial progressions

We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P ₁, …, P _k  ∈ Z[m] in one unknown m with P ₁(0) = … = P _k (0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with 1 \leqslant m \leqslant x^e1 \leqslant m \leqslant x^\varepsilon, such that x + P ₁(m), …, x + P _k (m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P _j  = (j − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

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.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Differential sobordination and Bazilevič functions

LetM(z)=z ⁿ +…,N(z)=z ⁿ +… be analytic in the unit disc Δ and let λ(z)=N(z)/zN′(z). The classical result of Sakaguchi-Libera shows that Re(M′(z)/N′(z))<0 implies Re(M(z)/N(z))>0 in Δ whenever Re(λ(z))>0 in Δ. This can be expressed in terms of differential subordination as follows: for anyp analytic in Δ, withp(0)=1,p(z)+λ(z)zp′(z)<1+z/1−z impliesp(z)<1+z/1−z, for Reλ(z)>0,z∈Δ. In this paper we determine different type of general conditions on λ(z),h(z) and ϕ(z) for which one hasp(z)+λ(z)zp′(z)<h(z) impliesp(z)<ϕ(z)<h(z) z∈Δ. Then we apply the above implication to obtain new theorems for some classes of normalized analytic funotions. In particular we give a sufficient condition for an analytic function to be starlike in Δ.     

The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains

The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1| (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α ＜ 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn ∑nj=1|zj|pj ＜ 1} (pj ＞ 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α ＜ 1 and some suitable constants βj,γj,pj.     

Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT _α=Σ _j=1 ^∞ a _j T ^j , where {a _j } are the coefficients in the power series expansion (1-t)^α=1-Σ _j=1 ^∞ a _j t ^j in the open unit disk, which satisfya _j >0 anda _j >0 and Σ _j=1 ^∞ a _j =1. The operator calculus justifies the notation(I−T) ^α :=I−T _α (e.g., (I−T _1/2)²=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) ^α x=y, i.e.,y is a coboundary forT _α . The fractional Poisson equation forT is the Poisson equation forT _α . We show that if(I−T)X is not closed, then(I−T) ^α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ _k=1 ^∞ T ^k y/k ^1-α converges in norm, and conclude that lim _n ‖(1/n ^1-α)Σ _k=1 ⁿ T ^k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL ₁ of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) ^α L ₁ for some 0<α<1, then the one-sided Hilbert transform Σ _k=1 ^∞ T ^k f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T) ^α L _p with α>1−1/p=1/q, then Σ _k=1 ^∞ T ^k f/k ^1/p converges a.e., and thus (1/n ^1/p ) Σ _k=1 ⁿ T ^k f converges a.e. to zero. Whenf∈(I−T) ^1/q L _p (the case α=1/q), we prove that (1/n ^1/p (logn)^1/q )Σ _k=1 ⁿ T ^k f converges a.e. to zero.     

Weyl’s theorem for algebraically wF(p, r, q) operators with p, r?>?0 and q?≥?1

If T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1 acting in an infinite-dimensional separable Hilbert space, then we prove that Weyl’s theorem holds for f(T) for any f ∈ Hol(σ(T)), where Hol(σ(T)) is the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is a wF(p, r, q) operator with p, r > 0 and q ≥ 1, then the a-Weyl’s theorem holds for f(T). In addition, if T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1, then we establish the spectral mapping theorems for the Weyl spectrum and for the essential approximate point spectrum of T for any f ∈ Hol(σ(T)), respectively. Finally, we examine the stability of Weyl’s theorem and the a-Weyl’s theorem under commutative perturbations by finite-rank operators.