Mathematics

In this paper, we investigate the Bohr radius for K -quasiregular sense-preserving harmonic mappings f=h+ g ¯ ¯ ¯ in the unit disk D such that the translated analytic part h(z)−h(0) is quasi-subordinate to some analytic function. The main aim of this article is to extend and to establish sharp versions of four recent theorems by Liu and Ponnusamy \cite{LP2019} and, in particular, we settle affirmatively the two conjectures proposed by them. Furthermore, we establish two refined versions of Bohr's inequalities and determine the Bohr radius for the derivatives of analytic functions associated with quasi-subordination.

In this paper we first consider another version of the Rogosinski inequality for analytic functions f(z)= ∑ ∞ n=0 a n z n in the unit disk |z|<1 , in which we replace the coefficients a n (n=0,1,…,N) of the power series by the derivatives f (n) (z)/n! (n=0,1,…,N) . Secondly, we obtain improved versions of the classical Bohr inequality and Bohr's inequality for the harmonic mappings of the form f=h+ g ¯ ¯ ¯ , where the analytic part h is bounded by 1 and that | g ′ (z)|≤k| h ′ (z)| in |z|<1 and for some k∈[0,1] .

Suppose w is a sense-preserving harmonic mapping of the unit disk D such that w(D)⊆D and w has a zero of order p≥1 at z=0 . In this paper, we first improve the Schwarz lemma for w , and then, we establish its boundary Schwarz lemma. Moreover, by using the automorphism of D , we further generalize this result.

We study the behaviors of the relative Bergman kernel metrics on holomorphic families of degenerating hyperelliptic Riemann surfaces and their Jacobian varieties. Near a node or cusp, we obtain precise asymptotic formulas with explicit coefficients. In general the Bergman kernels on a given cuspidal family do not always converge to that on the regular part of the limiting surface, which is different from the nodal case. It turns out that information on both the singularity and complex structure contributes to various asymptotic behaviors of the Bergman kernel. Our method involves the classical Taylor expansion for Abelian differentials and period matrices.

In this paper, we prove that the general ellipsoid D P is holomorphically homogeneous regular provided that it is a WB -domain. Then, the uniform lower bound for the squeezing function near a (P,r) -extreme point is also given.

For mapping with branching points that satisfy the inverse inequality of Poletsky, we obtained the results of their continuous boundary extension in terms of prime ends. Under certain conditions, the specified classes od mappings are also equicontinuous in the closure of a given domain.

In this paper we study the boundedness and compactness characterizations of the commutator of Cauchy type integrals C on a bounded strongly pseudoconvex domain D in C n with boundary bD satisfying the minimum regularity condition C 2 based on the recent result of Lanzani-Stein and Duong-Lacey-Li-Wick-Wu. We point out that in this setting the Cauchy type integral C is the sum of the essential part C ♯ which is a Calderón-Zygmund operator and a remainder R which is no longer a Calderón-Zygmund operator. We show that the commutator [b,C] is bounded on weighted Morrey space L p,κ v (bD) ( v∈ A p ,1<p<∞ ) if and only if b is in the BMO space on bD . Moreover, the commutator [b,C] is compact on weighted Morrey space L p,κ v (bD) ( v∈ A p ,1<p<∞ ) if and only if b is in the VMO space on bD .

We provide a boundedness criterion for the integral operator S ? on the fractional Fock-Sobolev space F s,2 ( C n ) , s?? , where S ? (introduced by Kehe Zhu) is given by S ? F(z):= ??C n F(w) e z??w ¯ ?(z??w ¯ )dλ(w) with ? in the Fock space F 2 ( C n ) and dλ(w):= ? ?�n e ?�|w | 2 dw the Gaussian measure on the complex space C n . This extends the recent result in Cao--Li--Shen--Wick--Yan. The main approach is to develop multipliers on the fractional Hermite-Sobolev space W s,2 H ( R n ) .

We present a Phragmén-Lindelöf type theorem with a flavor of Nevanlinna's theorem for subharmonic functions with frequent oscillations between zero and one. We use a technique inspired by a paper of Jones and Makarov.

The conditions on A , B , β and γ are obtained for an analytic function p defined on the open unit disc D and normalized by p(0)=1 to be subordinate to (1+Az)/(1+Bz) , ???�B<A?? when p(z)+z p ??(z)/(βp(z)+γ) is subordinate to e z . The conditions on these parameters are derived for the function p to be subordinate to 1+z ??????????or e z when p(z)+z p ??(z)/(βp(z)+γ) is subordinate to (1+Az)/(1+Bz) . The conditions on β and γ are determined for the function p to be subordinate to e z when p(z)+z p ??(z)/(βp(z)+γ) is subordinate to 1+z ??????????. Related result for the function p(z)+z p ??(z)/(βp(z)+γ) to be in the parabolic region bounded by the Rew=|w??| is investigated. Sufficient conditions for the Bernardi's integral operator to belong to the various subclasses of starlike functions are obtained as applications