Distortion theorems for functions regular and bounded in the disk

The paper refines and supplements some classical and modern distortion theorems for functions regular in the disk |z| < 1 and satisfying the condition |f(z)| < 1 for |z| < 1. Distortion bounds at points of the disk |z| < 1 and also on its boundary |z| = 1, provided that |f(z)| = 1, are considered both for univalent and multivalent mappings. Bibliography: 22 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 26–39.     

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.     

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 sum of coefficients of bounded univalent functions

We solve the maximal value problem for the functional in the class of functionsf(z)=z+a ₂z²+… that are holomorphic and univalent in the unit disk and satisfy the inequality |f(z)|<M. We prove that the Pick functions are extremal for this problem for sufficiently largeM whenever the set of indicesk ₁,…,k_m contains an even number. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 728–733, May, 1997. Translated by S. S. Anisov     

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.     

Entire functions, analytic continuation, and the fractional parts of a linear function

The main result of the paper is as follows.Theorem. Suppose that G(z) is an entire function satisfying the following conditions: 1) the Taylor coefficients of the function G(z) are nonnegative: 2) for some fixed C>0 and A>0 and for |z|>R₀, the following inequality holds:

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.     

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.     

The Region of Values of the System {f(z 1),..., f(z n )} in the Class of Typically Real Functions

The paper studies the region of values of the system {f(z₁), f(z₂),... , f(z_n)} in the class T of functions f(z) = z + a₂z² + ⋯ regular in the unit disk and satisfying the condition Im f(z) Im z > 0 for Im z ≠ 0. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 41–51.     

Characterizations of Hardy-Orlicz and Bergman-Orlicz spaces

Let and τ denote the invariant gradient and invariant measure on the unit ball B of ℂⁿ, respectively. Assume that f is a holomorphic function on B and ϕ ∈ C²(ℝ) is a nonnegative, nondecreasing, convex function. Then f belongs to the Hardy-Orlicz space H _ϕ(B>) if and only if

Estimates of Conformal Radius and Distortion Theorems for Univalent Functions

A simple proof of the recent result by E. G. Emel'yanov concerning the maximum of the conformal radius r(D,1) for a family of simply connected domains with a fixed value r(D,0) is given. A similar problem is solved for a family of convex domains. Exact estimates for functionals of the form are obtained for families of functions inverse to elements of the classes S and S_m, where S={f:f is regular and univalent in the disk {z:|z| < 1} and f(0)=f'(0)-1=0} and S_M= for . Bibliography: 7 titles.     

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.     

Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk \mathbbD{\mathbb{D}}. Each function f ? Co(a){f\in Co(\alpha)} maps the unit disk \mathbbD{\mathbb{D}} onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional (1-|z|²)( f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional (1-|z|²)(f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)} whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

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 Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

Sull'andamento delle funzioni maggioranti delle serie di potenze

Sunto Si considerano le serie di potenze f(z) = ∑a _n zⁿ convergenti (almeno) per | z |<1 e si studia l'andamento della funzione maggiorante ∑ | a_n | r^u, ponendolo in relazione con quello della media quadratica del modulo { ∑ | a_n |²r²ⁿ}^1/2 (che in ogni caso è minore del massimo modulo . Si stabiliscono le ? migliori costanti ? soltanto per qualcuno dei quesiti. Alla fine si esamina il caso in cui {∑ | a_n |²r²ⁿ} è convergente con ipotesi che tengono conto delle funzioni ? lentamente oscillanti ?. A Mauro Picone nel suo 70^mo compleanno.     

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.     

Boundary distortion and variation of the module under an extension of a doubly connected domain

Theorem 2

Let f(z) ∈ $\mathcal{F}(\rho ,r)$ , f(z) ≠ e^iα f(z;pr), α ∈ ?, and let ?(t) be a strictly convex monotone function of t>0. Then $$\int\limits_0^{2\pi } {\Phi (|f'(e^{i\theta } )|)d\theta< } \int\limits_0^{2\pi } {\Phi (|f'(e^{i\theta } ;\rho ,r)|)d\theta } $$ . The proof of this theorem is based on the Golusin-Komatu equation. If E is a continuum in the disk U_R={z:|z|<R}, then M (R, E) denotes the conformal module of the doubly connected component of U_R/E; let $\varepsilon (m) = \{ E:\overline U _r \subset E \subset U_1 , M(1,E) = M^{ - 1} \} $ .

Problem 3

Find the maximum of M(R, E), R>1, and the minimum of cap E over all E in ε(m). This problem was posed by V. V. Kozevnikov in a lecture to the Seminar on Geometric Function Theory at the Kuban University in 1980, and by D. Gaier (see [2]). The solution of this problem is given by the following theorem.

Theorem 3

Let $E^* = \underline U _m \cup [m,s]$ . If R>1; E, E^* ∈ ε(m) and E ≠ e^iα E*, α ∈ ?, then M(R, E)<M(R, E^*), capE^*<capE. A similar statement is also proved for continua lying in the half-plane. Bibliography: 7 titles.     

Some extremal problems for Pairs of functions with no common values

By the method of the moduli of families of curves one solves some extremal problems in the family of pairs {f₁, f₂} of functions f₁(z)=αz + ..., f₂(z)=βz^-1 + β⁰ +β¹z+..., with real coefficients, univalent, regular, resp. meromorphic in the circle Δ={∥z∥<1} and mapping onto nonoverlapping domains. As a special case the solution of a problem, posed by V. M. Miklyukov in Sib. Mat. Zh., Vol. 18, 1977, No. 5, pp. 1111–1124, is obtained. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 159–169, 1987.     

The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

Some sharp estimates for coefficients, distortion and the growth order are obtained for harmonic mappings $f ∈ TL^α_H$which are locally univalent harmonic mappings in the unit disk $\mathbb{D}:=\{z:|z| < 1\}$. Moreover, denoting the subclass $TS^α_H$ of the normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f ∈ TS^α_H,$ and their covering theorems.