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.     

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.     

Roots of the equationf (z)=αf (α) for the class of typically-real functions

Let T_r be the class of functionsf (z)=z+c₂z²+..., regular in the disk ¦z¦ <1, real on the diameter-1f (z) · Im z>0 in the remainder of the disk ¦z¦ <1. Let z _f be the solution off (z)= f (a) on Tr, where is any fixed complex number 0, 1, is any fixed real number, ¦¦< 1. We determine the region of values of the functional z_f on the class T_r. Variation formulas for Stieltjes integrals due to G.M. Goluzin are used.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 41–52, July, 1971.     

On Univalence of the Power Deformation z（f（z）/z）c

The authors mainly concern the set U _f of c ?? ? such that the power deformation $ z(\frac{{f(z)}} {z})^c $ is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a ₂ z ² + ?? in the unit disk. It is shown that U _f is a compact, polynomially convex subset of the complex plane ? unless f is the identity function. In particular, the interior of U _f is simply connected. This fact enables us to apply various versions of the ??-lemma for the holomorphic family $ z(\frac{{f(z)}} {z})^c $ of injections parametrized over the interior of U _f . The necessary or sufficient conditions for U _f to contain 0 or 1 as an interior point are also given.     

Carathéodory balls and norm balls of the domainH={(z1,z2)∈C2∶|z1|+|z2|<1}

LetH be the domain inC ² defined byH={Z=(z ₁,z ₂):║Z║₁=│z║₁│+│z║₂│<1}. LetC _H(z,w) be the Carathéodory distance ofH,z,w∈H. The Carathéodory ballB _C(z_C,α;H) with centerz _C,z_C∈H, and radius α, 0<α<1, is defined byB _c(z_C,α;H)={z∶C_H(z,z_C)<arc tanh α}. The norm ballB _N(z_N,r) with centerz _N,z_N∈H, and radiusr, 0<r<1-‖z _N‖₁, is defined byB _N(z_N,r)={z∶ ‖z−z_N‖₁<r}. Theorem:The only Carathéodory balls of H which are also norm balls are those with their center at the origin.     

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

Regions of Values of the Systems {f(z 1), f(z 2), f'(z 2)} and {f(z 1), f'(z 1), f'(z 1)} on the Class of Typically Real Functions

Let T be the class of functions f(z) = z + a ₂ z ² + . . . that are regular in the unit disk and satisfy the condition Im f(z) Im z > 0 for Im z 0, and let z ₁ and z ₂ be any distinct fixed points in the disk |z| < 1. For the systems of functionals mentioned in the title, the regions of values on T are studied. As a corollary, the regions of values of f'(z ₂) and f'(z ₁) on the subclasses of functions in T with fixed values f (z ₁), f (z ₂) and f (z ₁), f'(z ₁), respectively, are found. Bibliography: 7 titles.     

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.     

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.     

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.     

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.     

On the fourth and fifth coefficients of strongly starlike functions

For 0 < α ≤ 1, analytic functions f(z) = z + a₂z² + a₃z³ + … in the unit disk U are strongly starlike of order α if ¦arg {zf′ (z)/f(z)}¦ < πα / 2, z ∈ U. We find sharp estimates on the fourth and fifth coefficients of functions in this class.     

Uniqueness theorem of solutions for stochastic differential equation in the plane

LetM={M _z, z ∈ R ₊ ² } be a continuous square integrable martingale andA={A _z, z ∈ R ₊ ² be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane:dX _z=α(z, X_z)dM_z+β(z, X_z)dA_z, z∈R ₊ ² , X_z=Z_z, z∈∂R ₊ ² , whereR ₊ ² =[0, +∞)×[0,+∞) and ∂R ₊ ² is its boundary,Z is a continuous stochastic process on ∂R ₊ ² . We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2]. Supported by the National Science Foundation and the Postdoctoral Science Foundation of China     

De Bruijn’s question on the zeros of Fourier transforms

Letf(z) be a real entire function of genus 1^*, δ≥0, and suppose that for each ε>0, all but a finite number of the zeros off(z) lie in the strip |Imz| ≤δ+ε. Let λ be a positive constant such that . It is shown that for each ε>0, all but a finite number of the zeros of the entire function lie in the strip and if Δ² < 2λ, then all but a finite number of the zeros of e^−λD2 f(z) are real and simple. As a consequence, de Bruijn's question whether the functions e^γ t ²,λ>0, are strong universal factors is answered affirmatively. The authors wish to acknowledge the financial support of the Korea Research Foundation made in the program year of (1998–2000).     

On extremal quasiconformal extensions of conformal mappings

Letf(t, z)=z+tω(1/z) be schlicht for ⋎z⋎>1, ω(z) = Σ _n ^{= 0/∞} a _n z ⁿ ,t>0. The paper considers first-order estimates for the dilatation of extremal quasiconformal extensions off ast→0. This work was initiated during the Special Year in Complex Analysis at the Technion, and was supported in parts by the Samuel Neaman Fund, the Forschungsinstitut für Mathematik, ETH, Zürich, and the National Science Foundation.     

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.     

On generalized shift bases for the wiener disc algebra

LetW(D) denote the set of functionsf(z)=Σ _n=0 ^∞ A _n Z ⁿ a _nzⁿ for which Σ_n=0 ^∞|a _n |<+∞. Given any finite set lcub;f _i (z)rcub; _i=1 ⁿ inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f ₁(z)z ^kn ,f ₂(z)z ^kn+1, …,f _n (z)z ^(k+1)n−1rcub; _k=0 ^∞ is a basis forW(D) which is equivalent to the basis lcub;z ^m rcub; _m=0 ^∞ . (ii) The generalized shift sequence is complete inW(D), (iii) The function has no zero in |z|≦1, wherew=e ^2πiti /n.     

Region of Variability for Exponentially Convex Univalent Functions

For ${\alpha\in\mathbb C{\setminus}\{0\}}For a ? \mathbb C\{0}{\alpha\in\mathbb C{\setminus}\{0\}} let E(a){\mathcal{E}(\alpha)} denote the class of all univalent functions f in the unit disk \mathbbD{\mathbb{D}} and is given by f(z)=z+a₂z²+a₃z³+?{f(z)=z+a_2z^2+a_3z^3+\cdots}, satisfying

${\rm Re}\left (1+ \frac{zf'(z)}{f'(z)}+\alpha zf'(z)\right ) > 0 \quad {\rm in }\,{\mathbb D}.${\rm Re}\left (1+ \frac{zf'(z)}{f'(z)}+\alpha zf'(z)\right ) > 0 \quad {\rm in }\,{\mathbb D}.      20. 泛函方程组的解析解      刘新和 《高校应用数学学报(英文版)》2003,18(2):129-137 § 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…

泛函方程组的解析解

§ 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…