Structural formulas and value regions of functionals in certain classes of regular functions

We study the structural properties of the class M_k,λ,b(k≥2, 0≤λ≤1, b∈ℂ\{0}) of functions f(z)=z+ ... which are regular in |z|<1 and satisfy the conditions f(z)f′(z)z⁻¹≠0 and , where J(z)=λ(1+b⁻¹zf″(z)/f′(z)+(1−λ)(b⁻¹zf′(z)/f(z)+1−b⁻¹). The value regions of some functionals on this class are found. The case λ=1 was considered in our previous paper. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 55–60. Translated by O. A. Ivanov.     

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.     

Zeros in Tables of Characters for the Groups <Emphasis Type="Italic">S</Emphasis><Subscript>n</Subscript> and <Emphasis Type="Italic">A</Emphasis><Subscript>n</Subscript>. II

Let P(n) be the set of all partitions of a natural number n. In the representation theory of symmetric groups, for every partition α ∈ P(n), the partition h(α) ∈ P(n) is defined so as to produce a certain set of zeros in the character table for S_n. Previously, the analog f(α) of h(α) was obtained pointing out an extra set of zeros in the table mentioned. Namely, h(α) is greatest (under the lexicographic ordering ≤) of the partitions β of n such that χ^α(g_β) ≠ 0, and f(α) is greatest of the partitions γ of n that are opposite in sign to h(α) and are such that χ^α(g_γ) ≠ 0, where χ^α is an irreducible character of S_n, indexed by α, and g_β is an element in the conjugacy class of S_n, indexed by β. For α ∈ P(n), under some natural restrictions, here, we construct new partitions h′(α) and f′(α) of n possessing the following properties. (A) Let α ∈ P(n) and n ⩾ 3. Then h′(α) is identical is sign to h(α), χ^α(g_h′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of h(α), and h′(α) < γ < h(α). (B) Let α ∈ P(n), α ≠ α′, and n ⩾ 4. Then f′(α) is identical in sign to f(α), χ^α(g_f′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of f(α), and f′(α) < γ < f(α). The results obtained are then applied to study pairs of semiproportional irreducible characters in A_n. Supported by RFBR grant No. 04-01-00463. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 643–663, November–December, 2005.     

On integral means of star-like functions

We study univalent holomorphic functions in the unit diskU = {z: |z| < 1} of the formf(z)=z+∑ _n=2 ^∞ a _n z ⁿ that satisfy the condition Re zf’(z)/f(z) > α with α∈ [0, 1) inU. Some integral means of such funcions are estimated.     

Equivalences between bloch type spaces and BOM spaces on the hyperbolic domains inC

For an analytic function f on the hyperbolic domain Ω inC, the following conclusions are obtained: (i) f∈B(Ω)=BMO A(Ω,m) if and only ifRef∈B_h(Ω)=BMOH(Ω,m). (ii) QB_h(Ω)=B_h(Ω)(BMOH _n(Ω,m)=BMOH(Ω,m)) if and only ifC(Ω)=inf{λ_Ω(z)·δ_Ω(z):z∈Ω}>0. Also, some applications to automorphic functions are considered. This research was supported by the Doctoral Program Foundation of Institute of Higher Education.     

BMO-scale of distribution on ℝ n

Let S′ be the class of tempered distributions. For ƒ ∈ S′ we denote by J ^−α ƒ the Bessel potential of ƒ of order α. We prove that if J ^−α ƒ ∈ BMO, then for any λ ∈ (0, 1), J ^−α (f)_λ ∈ BMO, where (f)_λ = λ⁻ⁿ f(φ(λ⁻¹)), φ ∈ S. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order α > 0 belongs to the VMO space.     

The value regions of initial coefficients in a certain class of meromorphic functions

Let M_k,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+a_o+a₁z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition where J_λ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class M_k,λ we consider sorne coefficient problems and problems concerning distortion theorems. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 91–96. Translated by N. Yu. Netsvetaev.     

On convergence of a generalized Pál interpolation process

Let f∈C _[−1,1] ^″ (r≥1) and R_n(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions R_n(f,α,β,x_k)=f(x_k),R_n ^′(f,α,β,x_k)=f′(x_k)(k=1,2,…,n), where {x_k} are the roots of n-th Jacobi polynomial P_n(α,β,x),α,β>−1 and {x _k ^″ } are the roots of (1−x²)P_n″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.     

On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators

^m σ/|α|=|β|=m D^α(a _αβD^β) is a higher order degenerate-elliptic operator on L²(R^N) and V ∈ L¹(R^N) we show in particular that s(λ) > 0 for all λ > 0 provided that ∫_RN V(x)dx≥ 0, V not equivalent to 0 and N ≤ 2m.     

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 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

Differential subordinations concerning starlike functions

Denote byS ^* (⌕), (0≤⌕<1), the family consisting of functionsf(z)=z+a ₂z²+...+a_nzⁿ+... that are analytic and starlike of order ⌕, in the unit disc ⋎z⋎<1. In the present article among other things, with very simple conditions on μ, ⌕ andh(z) we prove the f’(z) (f(z)/z)^μ−1<h(z) implies f∈S^*(⌕). Our results in this direction then admit new applications in the study of univalent functions. In many cases these results considerably extend the earlier works of Miller and Mocanu [6] and others.     

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.     

Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.     

Boundary functions on a bounded balanced domain

We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.      

Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville problem

A continuous extension of the de la Vallée Poussin means

Among the many interesting results of their 1958 paper, G. Pólya and I. J. Schoenberg studied the de la Vallée Poussin means of analytic functions. These are polynomial approximations of a given analytic function on the unit disk obtained by taking Hadamard products of the functionf with certain polynomialsV _n (z), wheren is the degree of the polynomial. The polynomial approximationsV _n *f converge locally uniformly tof asn→∞. In this paper, we define a subordination chainV _λ (z),γ>0, |z|<1, of convex mappings of the disk that for integer values is the same as the previously definedV _n (z). Iff is a conformal mapping of the diskD onto a convex domain, thenV _λ *f→f locally uniformly as λ→∞, and in fact when λ₂ > λ₁. We also consider Hadamard products of theV _λ with complex-valued harmonic mappings of the disk. This work was supported by the Volkswagen Stiftung (RiP-program at Oberwolfach). S. R. received partial support also from INTAS (Project 99-00089) and the German-Israeli Foundation (grant G-643-117.6/1999).     

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.