Mathematics

In this article, we investigate the (big) Hankel operators H f on Hardy spaces of strongly pseudoconvex domains with smooth boundaries in C n . We also give a necessary and sufficient condition for boundedness of the Hankel operator H f on the Hardy space of the unit disc, which is new in the setting of one variable.

We study analogues of well-known relationships between Muckenhoupt weights and BMO in the setting of Bekollé-Bonami weights. For Bekollé-Bonami weights of bounded hyperbolic oscillation, we provide distance formulas of Garnett and Jones-type, in the context of BMO on the unit disc and hyperbolic Lipschitz functions. This leads to a characterization of all weights in this class, for which any power of the weight is a Bekollé-Bonami weight, which in particular reveals an intimate connection between Bekollé-Bonami weights and Bloch functions. On the open problem of characterizing the closure of bounded analytic functions in the Bloch space, we provide a counter-example to a related recent conjecture. This shed light into the difficulty of preserving harmonicity in approximation problems in norms equivalent to the Bloch norm. Finally, we apply our results to study certain spectral properties of Cesaró operators.

We prove that the family F C (D) of all meromorphic functions f on a domain D⊆C with the property that the spherical area of the image domain f(D) is uniformly bounded by Cπ is quasi--normal of order ≤C . We also discuss the close relations between this result and the well--known work of Brézis and Merle on blow--up solutions of Liouville's equation. These results are completely in the spirit of Gromov's compactness theorem, as pointed out at the end of the paper.

Inspired by the idea of blurring the exponential function, we define blurred variants of the j -function and its derivatives, where blurring is given by the action of a subgroup of GL 2 (C) . For a dense subgroup (in the complex topology) we prove an Existential Closedness theorem which states that all systems of equations in terms of the corresponding blurred j with derivatives have complex solutions, except where there is a functional transcendence reason why they should not. For the j -function without derivatives we prove a stronger theorem, namely, Existential Closedness for j blurred by the action of a subgroup which is dense in GL + 2 (R) , but not necessarily in GL 2 (C) . We also show that for a suitably chosen countable algebraically closed subfield C⊆C , the complex field augmented with a predicate for the blurring of the j -function by GL 2 (C) is model theoretically tame, in particular, ω -stable and quasiminimal.

In this article, we determine sharp Bohr-type radii for certain complex integral operators defined on a set of bounded analytic functions in the unit disk.

The Bohr phenomenon for analytic functions of the form f(z)= ∑ ∞ n=0 a n z n , first introduced by Harald Bohr in 1914, deals with finding the largest radius r f , 0< r f <1 , such that the inequality ∑ ∞ n=0 | a n z n |≤1 holds whenever the inequality |f(z)|≤1 holds in the unit disk D={z∈C:|z|<1} . The exact value of this largest radius known as Bohr radius, which has been established to be r f =1/3 . The Bohr phenomenon \cite{Abu-2010} for harmonic functions f of the form f(z)=h(z)+ g(z) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ , where h(z)= ∑ ∞ n=0 a n z n and g(z)= ∑ ∞ n=1 b n z n is to find the largest radius r f , 0< r f <1 such that ∑ n=1 ∞ (| a n |+| b n |)|z | n ≤d(f(0),∂f(D)) holds for |z|≤ r f , here d(f(0),∂f(D)) denotes the Euclidean distance between f(0) and the boundary of f(D) . In this paper, we investigate the Bohr radius for several classes of harmonic functions in the unit disk D.

We say that a class B of analytic functions f of the form f(z)= ∑ ∞ n=0 a n z n in the unit disk D:={z∈C:|z|<1} satisfies a Bohr phenomenon if for the largest radius R f <1 , the following inequality ∑ n=1 ∞ | a n z n |≤d(f(0),∂f(D)) holds for |z|=r≤ R f and for all functions f∈B . The largest radius R f is called Bohr radius for the class B . In this article, we obtain Bohr radius for certain subclasses of close-to-convex analytic functions. We establish the Bohr phenomenon for certain analytic classes S ∗ c (ϕ), C c (ϕ), C ∗ s (ϕ), K s (ϕ) . Using Bohr phenomenon for subordination classes \cite[Lemma 1]{bhowmik-2018}, we obtain some radius R f such that Bohr phenomenon for these classes holds for |z|=r≤ R f . Generally, in this case R f need not be sharp, but we show that under some additional conditions on ϕ , the radius R f becomes sharp bound. As a consequence of these results, we obtain several interesting corollaries on Bohr phenomenon for the aforesaid classes.

The purpose of this article is to study Bohr inequalities involving the absolute values of the coefficients of an operator valued function. To be more specific, we establish an operator valued analogue of a classical result regarding the Bohr phenomenon for scalar valued functions with fixed initial coefficient. Apart from that, operator valued versions of other related and well known results are obtained.

We say that a class F consisting of analytic functions f(z)= ∑ ∞ n=0 a n z n in the unit disk D:={z∈C:|z|<1} satisfies a Bohr phenomenon if there exists r f ∈(0,1) such that ∑ n=1 ∞ | a n z n |≤d(f(0),∂f(D)) for every function f∈F and |z|=r≤ r f , where d is the Euclidean distance. The largest radius r f is the Bohr radius for the class F . In this paper, we establish the Bohr phenomenon for the classes consisting of Ma-Minda type starlike functions and Ma-Minda type convex functions as well as for the class of starlike functions with respect to a boundary point.

Recently, there has been a number of good deal of research on the Bohr's phenomenon in various setting including a refined formulation of his classical version of the inequality. Among them, in \cite{PaulPopeSingh-02-10} the authors considered the cases in which the above functions have a multiple zero at the origin. In this article, we present a refined version of Bohr's inequality for these cases and give a partial answer to a question from \cite{PaulPopeSingh-02-10} for the revised setting.