9,471 research outputs found
Distortion theorems for Bloch functions
No full textThis paper has a similar theme. We show that there is an analogous subordination theorem (Theorem 1) for normalized (not necessarily locally univalent) Bloch functions. This subordination result enables us to obtain some known results from a unified perspective and also leads to new results. Let us introduce some notation and terminology. The unit disk in the complex plane is denoted by D . For a function f holomorphic on D the Bloch seminorm is given b
Some characterizations of semi-Bloch functions
No full textA function
f
f
analytic in the unit disk is called a semi-Bloch function if, for each complex number
λ
\lambda
, the function
g
λ
(
z
)
=
exp
(
λ
f
(
z
)
)
{g_\lambda }(z) = \exp (\lambda f(z))
is a normal function. We give both an analytic and a geometric characterization of semi-Bloch functions, together with some examples to show that semi-Bloch functions are not closed under either addition or multiplication.</p
Semi-Bloch Functions in Several Complex Variables
No full textLet M be an n-dimensional complex manifold. A holomorphic function f:M→C is said to be semi-Bloch if for every λ∈C the function (Formula presented.) is normal on M. We characterize semi-Bloch functions on infinitesimally Kobayashi non-degenerate M in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.</p
On harmonic Bloch-type mappings
No full textLet f be a complex-valued harmonicmapping defined in the unit disk D. We introduce the following notion: we say that f is a Bloch-type function if its Jacobian satisfies This gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which, roughly speaking, states that for. analytic log. is Bloch if and only if. is univalent
On harmonic Bloch-type mappings
No full textLet f be a complex-valued harmonic mapping defined in the unit disk (Formula presented.). We introduce the following notion: we say that f is a Bloch-type function if its Jacobian satisfies (Formula presented.) This gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which, roughly speaking, states that for ϕ analytic ϕ' is Bloch if and only if ϕ is univalen
Semi-Bloch functions in several complex variables
No full textLet be an -dimensional complex manifold. A holomorphic function is said to be semi-Bloch if for every the function is normal on . We characterise Semi-Bloch functions on infinitesimally Kobayashi non-degenerate in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.</p
Inner functions in the hyperbolic little Bloch class
No full textAbstract. An analytic function ϕ mapping the unit disk into itself is said to belong to the hyperbolic little Bloch class if the ratio (1−|z | 2)|ϕ ′ (z)|/(1−|ϕ(z) | 2) converges to 0 as |z | → 1, while ϕ is in the little Bloch space if just the numerator of this expression converges to zero. Several constructions of inner functions in the little Bloch space have recently appeared. In this paper we construct a singular measure on the unit circle such that the associated singular inner function is in the hyperbolic little Bloch class. 1
On harmonic Bloch-type mappings
No full text<p>Let f be a complex-valued harmonic mapping defined in the unit disk D. We introduce the following notion: we say that f is a Bloch-type function if its Jacobian satisfies that (1-|z|^2)|J_f(z)|^{1/2} is uniformly bounded in D. This gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which states that an analytic ϕ is Bloch if and only if there exists c>0 and a univalent ψ such that ϕ=c*log(ψ').</p>
On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives
No full textAbstract
In this article, we study the Beilinson–Bloch–Kato conjecture for motives associated to Rankin–Selberg products of conjugate self-dual automorphic representations, within the framework of the Gan–Gross–Prasad conjecture. We show that if the central critical value of the Rankin–Selberg L-function does not vanish, then the Bloch–Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch–Kato Selmer group constructed from a certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin–Selberg L-function, then the Bloch–Kato Selmer group is of rank one
On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives
No full text212 page; comments are welcomeIn this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg -function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin-Selberg -function, then the Bloch-Kato Selmer group is of rank one
- …
