On power deformations of univalent functions

For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) ? 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f _c (z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator \({f \mapsto f_c}\) maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.     

Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

For ?1≤B<A≤1, let \(\mathcal {S}^{*}(A,B)\) denote the class of normalized analytic functions \(f(z)= z+{\sum }_{n=2}^{\infty }a_{n} z^{n}\) in |z|<1 which satisfy the subordination relation z f ^′(z)/f(z)?(1 + A z)/(1 + B z) and Σ^?(A,B) be the corresponding class of meromorphic functions in |z|>1. For \(f\in \mathcal {S}^{*}(A,B)\) and λ>0, we shall estimate the absolute value of the Taylor coefficients a _n (?λ,f) of the analytic function (f(z)/z)^?λ . Using this we shall determine the coefficient estimate for inverses of functions in the classes \(\mathcal {S}^{*}(A,B)\) and Σ^?(A,B).     

Fekete-Szegö problem for close-to-convex functions with respect to a certain convex function dependent on a real parameter

Given α ∈ [0, 1], let h _α (z):= z/(1 - αz), z ∈ D:= {z ∈ D: |z| < 1}. An analytic standardly normalized function f in D is called close-to-convex with respect to h _α if there exists δ ∈ (-π/2, π/2) such that Re{e^iδ zf′(z)/h _α (z)} > 0, z ∈ D. For the class ? (h _α ) of all close-to-convex functions with respect to h _α , the Fekete-Szegö problem is studied.     

Some Properties of a Singular Integral Equation with Non-Carleman Shift

We study the distribution of roots of the function W(z) = 1?λl^?z cot(πz), z ∈ C, where l ∈ (0, 1) and λ ∈ C are parameters. The results are applied to the problem on the solvability and number of solutions of a singular integral equation in various function spaces.     

Quasisymmetric geometry of the Julia sets of McMullen maps

We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z^m + λ/z^?, where λ ∈ ? {0} and ? and m are positive integers satisfying 1/?+1/m < 1. If the free critical points of f_λ are escaped to the infinity, we prove that the Julia set J_λ of f_λ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet (which is also standard in some sense). If the free critical points are not escaped, we give a suffcient condition on λ such that J_λ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.     

On the equiconvergence rate of trigonometric series expansions and eigenfunction expansions for the Sturm-Liouville operator with a distributional potential

We study the Sturm-Liouville operator L = ?d ²/dx ² + q(x) in the space L ₂[0, π] with the Dirichlet boundary conditions. We assume that the potential has the form q(x) = u′(x), u ∈ W ₂ ^θ [0, π], 0 < θ < 1/2. We consider the problem on the uniform (on the entire interval [0, π]) equiconvergence of the expansion of a function f(x) in a series in the system of root functions of the operator L with its Fourier expansion in the system of sines. We show that if the antiderivative u(x) of the potential belongs to any of the spaces W ₂ ^θ [0, π], 0 < θ < 1/2, then the equiconvergence rate can be estimated uniformly over the ball u(x) ∈ B _R = {v(x) ∈ W ₂ ^θ [0, π] | ∥v∥_{W 2} ^θ ≤ R} for any function f(x) ∈ L ₂[0, π].     

Order of magnitude of multiple Fourier coefficients of functions of bounded <Emphasis Type="Italic">p</Emphasis>-variation

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus \(\mathbb{T}^n \):= [0, 2π) ⁿ , let \(\hat f\)(k) denote the Fourier coefficient of f, where k = (k ₁, … k _n ) ∈ ? ⁿ . In this paper, defining the notion of bounded p-variation (p ≧ 1) for a function from [0, 2π] ⁿ to ? in two diffierent ways, the order of magnitude of Fourier coefficients of such functions is studied. As far as the order of magnitude is concerned, our results with p = 1 give the results of Móricz [5] and Fülöp and Móricz [3].     

Hyperbolic Chevalley groups on ℂ<Superscript>2</Superscript>

Let Γ ? U (1, 1) be the subgroup generated by the complex reflections. Suppose that Γ acts discretely on the domain K = {(z ₁, z ₂) ∈ ?² ||z ₁|² ? |z ₂|² < 0} and that the projective group PΓ acts on the unit disk B = {|z ₁/z ₂| < 1} as a Fuchsian group of signature (n ₁, ..., n _s ), s ? 3, n _i ? 2. For such groups, we prove a Chevalley type theorem, i.e., find a necessary and sufficient condition for the quotient space K/Γ to be isomorphic to ?² ? {0}.     

Solution of Functional Equations Related to Elliptic Functions

Functional equations of the form f(x + y)g(x ? y) = Σ _j=1 ⁿ α _j (x)β _j (y) as well as of the form f₁(x + z)f₂(y + z)f₃(x + y ? z) = Σ _j=1 ^m φ _j (x, y)ψ _j (z) are solved for unknown entire functions f, g,α _j , β _j : ? → ? and f₁, f₂, f₃, ψ _j : ? → ?, φ _j : ?² → ? in the cases of n = 3 and m = 4.     

Elliptic function of level 4

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u² ? v²)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uA(v)² ? vA(u)²), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A₁, and B″(0) = 2B₂ the following relation holds: (2B(u) + 3A₁u)² = 4A(u)³ ? (3A₁² ? 8B₂)u²A(u)². To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.     

A pure smoothness condition for radó’s theorem for α-analytic functions

Let \(\Omega \subset {{\Bbb C}^n}\) be a bounded, simply connected ?-convex domain. Let α ∈ ?₊ⁿ and let f be a function on Ω which is separately \({C^{2{\alpha _j} - 1}}\)-smooth with respect to z_j (by which we mean jointly \({C^{2{\alpha _j} - 1}}\)-smooth with respect to Rez_j, Imz_j). If f is α-analytic on Ω\f^?1(0), then f is α-analytic on Ω. The result is well-known for the case α_i = 1, 1 ? i ? n, even when f a priori is only known to be continuous.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

Modified boundary element method for problems on oscillations of flat membranes

In the space L ² of real-valued measurable 2π-periodic functions that are square summable on the period [0, 2π], the Jackson-Stechkin inequality

$$E_n (f) \leqslant \mathcal{K}_n (\delta ,\omega )\omega (\delta ,f), f \in L^2 $$

, is considered, where E _n (f) is the value of the best approximation of the function f by trigonometric polynomials of order at most n and ω(δ, f) is the modulus of continuity of the function f in L ² of order 1 or 2. The value

$$\mathcal{K}_n (\delta ,\omega ) = \sup \left\{ {\frac{{E_n (f)}}{{\omega (\delta ,f)}}:f \in L^2 } \right\}$$

is found at the points δ = 2π/m (where m ∈ ?) for m ≥ 3n ² + 2 and ω = ω ₁ as well as for m ≥ 11n ⁴/3 ? 1 and ω = ω ₂.

     

Structure of multiple solutions for nonlinear differential equations

Based on the eigensystem {λ_j,φ_j} of -Δ, the multiple solutions for nonlinear problem Δu + f(u) = 0 in Ω,u = 0 on ?Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u ³, u ², (u - p), u ²(u ² - p), are completed and some conjectures are presented.     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

A four-point criterion for the Möbius property of a homeomorphism of plane domains

An ordered quadruple of pairwise distinct points T = {z ₁, z ₂, z ₃, z ₄} ? C is called regular whenever z ₂ and z ₄ lie at the opposite sides of the line through z ₁ and z ₃. Consider Φ(T) = ∠z ₁ z ₂ z ₃ + ∠z ₁ z ₄ z ₃ (the angles are undirected) as some geometric characteristic of a regular tetrad. We prove the following theorem: For every fixed α ∈ (0, 2π) the Möbius property of a homeomorphism f: D → D* of domains in C is equivalent to the requirement that each regular tetrad T ? D with Φ(T) = α whose image fT is also a regular tetrad satisfies Φ(fT) = α. In 1994 Haruki and Rassias established this criterion for the Möbius property only in the class of univalent analytic functions f(z).     

Equiconvergence of the trigonometric Fourier series and the expansion in eigenfunctions of the Sturm-Liouville operator with a distribution potential

We consider the Sturm-Liouville operator L = ?d ²/dx ² + q(x) with the Dirichlet boundary conditions in the space L ₂[0, π] under the assumption that the potential q(x) belongs to W ₂ ^?1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L ₁. We also consider the case of potentials belonging to the scale of Sobolev spaces W ₂ ^?θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W ₂ ^θ [0, π] with 0 < θ < 1/2, then, for any function in the space L ₂[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence.     

On the existence of strips inside domains convex in one direction

A plane domain Ω is convex in the positive direction if for every ω ∈ Ω, the entire half-line {ω + t: t ≥ 0} is contained in Ω. Suppose that h maps the unit disk onto such a domain Ω with the normalization h(0) = 0 and lim_t→∞h^?1(h(z) + t) = 1. We show that if ∠lim_z→?1 Re h(z) = ?∞ and ∠lim_z→?1(1 + z)h′(z) = ν ∈ (0, +∞), then Ω contains a maximal horizontal strip of width πν. We also prove a converse statement. These results provide a solution to a problem posed by Elin and Shoikhet in connection with semigroups of holomorphic functions.     

Cocycle rigidity of abelian partially hyperbolic actions

Suppose G is a higher-rank connected semisimple Lie group with finite center and without compact factors. Let G = G or G = G ? V, where V is a finite-dimensional vector space V. For any unitary representation (π,H) of G, we study the twisted cohomological equation π(a)f ? λf = g for partially hyperbolic element a ∈ G and λ ∈ U(1), as well as the twisted cocycle equation π(a₁)f ? λ1f = π(a₂)g ? λ₂g for commuting partially hyperbolic elements a₁, a₂ ∈ G. We characterize the obstructions to solving these equations, construct smooth solutions and obtain tame Sobolev estimates for the solutions. These results can be extended to partially hyperbolic flows in parallel.As an application, we prove cocycle rigidity for any abelian higher-rank partially hyperbolic algebraic actions. This is the first paper exploring rigidity properties of partially hyperbolic that the hyperbolic directions don’t generate the whole tangent space. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for these actions in future works.