Bloch Constants Research Articles

Improved Bloch and Landau constants for meromorphic functions

AbstractLet ${\mathbb D}$ be the open unit disk, and let $\mathcal {A}(p)$ be the class of functions f that are holomorphic in ${\mathbb D}\backslash \{p\}$ with a simple pole at $z=p\in (0,1)$ , and $f'(0)\neq 0$ . In this article, we significantly improve lower bounds of the Bloch and the Landau constants for functions in ${\mathcal A}(p)$ which were obtained in Bhowmik and Sen (2023, Monatshefte für Mathematik, 201, 359–373) and conjecture on the exact values of such constants.

Landau and Bloch constants for meromorphic functions

Landau and Bloch constants for meromorphic functions

Precise values of the Bloch constants of certain log-p-harmonic mappings

The aim of this article is twofold. One aim is to establish the precise forms of Landau-Bloch type theorems for certain polyharmonic mappings in the unit disk by applying a geometric method. The other is to obtain the precise values of Bloch constants for certain log-p-harmonic mappings. These results improve upon the corresponding results given in Bai et al. (Complex Anal. Oper. Theory, 13(2): 321–340, 2019).

Landau Theorem For Planar Harmonic Mappings

Let \(w=P[F]\) be a harmonic mapping of the unit disk \({\mathbb {D}}\) with the boundary function \(F\). By using Poisson formula, we obtain some better estimates on Bloch constants for planar harmonic mappings.

LANDAU’S THEOREM AND MARDEN CONSTANT FOR HARMONIC ν-BLOCH MAPPINGS

AbstractOur main aim is to investigate the properties of harmonic ν-Bloch mappings. Firstly, we establish coefficient estimates and a Landau theorem for harmonic ν-Bloch mappings, which are generalizations of the corresponding results in Bonk et al. [‘Distortion theorems for Bloch functions’, Pacific. J. Math.179 (1997), 241–262] and Chen et al. [‘Bloch constants for planar harmonic mappings’, Proc. Amer. Math. Soc.128 (2000), 3231–3240]. Secondly, we obtain an improved Landau theorem for bounded harmonic mappings. Finally, we obtain a Marden constant for harmonic mappings.

Distortion theorems for Bloch mappings on the unit polydisc D n

In this article, we establish distortion theorems for some various subfamilies of Bloch mappings defined in the unit polydisc D n with critical points, which extend the results of Liu and Minda to higher dimensions. We obtain lower bounds of |det ( f'( z))| and ℝ det ( f'( z)) for Bloch mapping f. As an application, some lower and upper bounds of Bloch constants for the subfamilies of holomorphic mappings are given.

Landau’s theorem for certain biharmonic mappings

In this paper, we show the existence of Landau and Bloch constants for biharmonic mappings of the form L ( F ) . Here L represents the linear complex operator L = z ∂ ∂ z - z ¯ ∂ ∂ z ¯ defined on the class of complex-valued C 1 functions in the plane, and F belongs to the class of biharmonic mappings of the form F ( z ) = | z | 2 G ( z ) + K ( z ) ( | z | < 1 ) , where G and K are harmonic.

Estimates on Bloch constants for planar harmonic mappings

The Bloch constants for quasiregular harmonic mappings and open planar harmonic mappings are considered. Better estimates are obtained. The results, presented in this paper, improve the one made by Chen et al. and Grigoryan.

Bloch constant of holomorphic mappings on the unit polydisk of ℂ n

In this paper, we give a definition of Bloch mappings defined in the unit polydisk Dn, which generalizes the concept of Bloch functions defined in the unit disk D. It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables. We shall establish the corresponding distortion theorems for subfamilies β(K) and βloc(K) of Bloch mappings defined in the polydisk Dn, which extend the distortion theorems of Liu and Minda to higher dimensions. As an application, we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloch mappings defined in Dn. In particular, our results reduce to the classical results of Ahlfors and Landau when n = 1.

Bloch Constant of Holomorphic Mappings on the Unit Ball of ℂ n *

In this paper, the authors establish distortion theorems for various subfamilies H k (\(\Bbb {B}\)) of holomorphic mappings defined in the unit ball in ℂ n with critical points, where k is any positive integer. In particular, the distortion theorem for locally biholomorphic mappings is obtained when k tends to +∞. These distortion theorems give lower bounds on | det f′(z)| and Re det f′(z). As an application of these distortion theorems, the authors give lower and upper bounds of Bloch constants for the subfamilies β k (M) of holomorphic mappings. Moreover, these distortion theorems are sharp. When \(\Bbb {B}\) is the unit disk in ℂ, these theorems reduce to the results of Liu and Minda. A new distortion result of Re det f′(z) for locally biholomorphic mappings is also obtained.

Estimates on Bloch constants for planar harmonic mappings

The Schwarz lemma and Bloch constants for planar bounded harmonic mappings are considered. Sharper form and better estimates are obtained. Our results improve the one made by Dorff and Nowak as well as by Chen, Gauthier and Hengartner.

Bloch functions and Bloch constants

In this paper, the lower bounds of Bloch constants B n, for functions with multiplicity at least n, are improved by showing B i> n(n+2) 2(n+1) + 3×10 −15 n 3

Julia’s lemma and bloch constants

The classical Julia's lemma is improved, branch points of Bloch functions are investigated, and new lower bounds of Bloch constants for functions with branch points are obtained.

Magnetic properties of SiO2-coated Fe nanoparticles

SiO 2 -coated Fe nanoparticles were synthesized using a wet chemical method, and their structural and magnetic properties were studied. The SiO2 material was in an amorphous state. The Fe nanoparticles were in a bcc state and contained an inner ferrihydrite core whose size decreased with increasing calcination temperature. The nanoparticles were basically in the ferromagnetic state. Their saturation magnetization increased with increasing calcination temperature, whereas their coercivity decreased with increasing calcination temperature. Different from bulk Fe, the nanoparticles exhibited strong temperature-dependent magnetic behaviors. The Bloch exponent fell from 1.5 to smaller values and decreased with increasing ferrihydrite content, while the Bloch constants were much bigger than that for bulk and increased significantly with ferrihydrite content. The value of coercivity decreased notably with increasing temperature. The exchange anisotropy arising from the exchange coupling across the Fe/ferrihydrite interfaces was examined and was used to interpret the observed temperature behaviors.

Bloch constants in several variables

We give lower estimates for Bloch's constant for quasiregular holomorphic mappings. A holomorphic mapping of the unit ball Bn into Cn is K-quasiregular if it maps infinitesimal spheres to infinitesimal ellipsoids whose major axes are less than or equal to K times their minor axes. We show that if f is a K-quasiregular holomorphic mapping with the normalization det f'(O) = 1, then the image f(Bn) contains a schlicht ball of radius at least 1/12K1-1/n. This result is best possible in terms of powers of K. Also, we extend to several variables an analogous result of Landau for bounded holomorphic functions in the unit disk.

Bloch constants of bounded symmetric domains

Let D 1 D_{1} and D 2 D_{2} be two irreducible bounded symmetric domains in the complex spaces V 1 V_{1} and V 2 V_{2} respectively. Let E E be the Euclidean metric on V 2 V_{2} and h h the Bergman metric on V 1 V_{1} . The Bloch constant b ( D 1 , D 2 ) b(D_{1}, D_{2}) is defined to be the supremum of E ( f ′ ( z ) x , f ′ ( z ) x ) 1 2 / h z ( x , x ) 1 / 2 E(f^{\prime }(z)x, f^{\prime }(z)x)^{\frac {1}{2}}/h_{z}(x, x)^{1/2} , taken over all the holomorphic functions f : D 1 → D 2 f: D_{1}\to D_{2} and z ∈ D 1 z\in D_{1} , and nonzero vectors x ∈ V 1 x\in V_{1} . We find the constants for all the irreducible bounded symmetric domains D 1 D_{1} and D 2 D_{2} . As a special case we answer an open question of Cohen and Colonna.

Bloch constants in one and several variables

We give covering theorems in one variable for holomorphic functions on the unit disc with A -fold symmetry. In the case of convex maps we give a generalization , shown to us by D. Minda, to the case where «2 — — &fc — 0. In several variables we determine the Bloch constant (equivalently the Koebe constant) for convex maps of Bn with A -fold symmetry, k > 2. We also estimate and in some cases compute the Bloch constant for starlike maps of Bn with A -fold symmetry. We compare the Bloch constant with the Koebe constant for such maps and determine values of n and k for which equality holds. 1. Intrduction. This paper is the result of an attempt to answer the following Question. If F : Bn —>> Cn is a biholomorphic map of the unit ball onto a convex domain such that dF(0) — /, must F(Bn) contain a ball of radius

Distortion theorems for Bloch functions

We establish various distortion theorems for both normalized locally schlicht Bloch functions and normalized Bloch function with branch points. These distortion theorems give lower bounds on either | f ′ ( z ) | |f\prime (z)| or Re ⁡ f ′ ( z ) \operatorname {Re} f\prime (z) ; most of our distortion theorems are sharp and all extremal functions identified. The main tools used in establishing these distortion theorems are the classical form of Julia’s Lemma and a new version of Julia’s Lemma that applies to certain multiple-valued analytic functions. As applications of these distortion theorems, we obtain known lower bounds for various Bloch constants and also establish improved lower bounds on a number of Marden constants for Bloch, normal and Yosida functions.

The hyperbolic metirc and bloch constants for spherically convex regions

Let Ω denote a simply connected region on the Riemann sphere P such that Ω is convex relative to spherical geometry and Ω ≠ P. The quantity (1 +|z|2λΩ(z) is called the spherical density of the hyperbolic metric λΩ(z)|dz|on Ω. Two sharp lower bounds for the spherical density are obtained. First (1 +|z|2)λΩ(z)⩾ cosec with equality if and only if Ω is a hemisphere, where denotes the spherical distance from z to . Second, if for all ζ∈Ω then λΩ(z)⩾π/4O These bounds lead to covering theorems for the class K sα of all univalent functions f in the unit disk D such that and f(D) is spherically convex. The Koebe set f ∈ K s(α)) is the spherical disk about the origin with spherical radius (1 2) aresin α while the spherical Bloch constant for the family K s(α)is απ4. For both covering results all extremal functions are identified.

Bloch Constants for Meromorphic Functions Near an Isolated Singularity

Suppose $f$ is meromorphic in a punctured neighborhood of the origin and has an essential singularity at the origin. Given any $\varepsilon > 0$ we show that the Riemann surface of $f$ contains an unramified disk of spherical radius $\pi /3 - \varepsilon$. The number $\pi /3$ can be replaced by $\pi /2$ if $f$ is locally schlicht and this value is best possible. If $f$ is actually holomorphic, then the Riemann surface of $f$ contains arbitrarily large unramified euclidean disks. These results generalize theorems of Valiron and Ahlfors dealing with holomorphic and meromorphic functions, respectively, on the complex plane which have an essential singularity at infinity.