Mathematics

We give a general formula for the C− transfinite diameter δ C (K) of a compact set K⊂ C 2 which is a product of univariate compacta where C⊂( R + ) 2 is a convex body. Along the way we prove a Rumely type formula relating δ C (K) and the C− Robin function ρ V C,K of the C− extremal plurisubharmonic function V C,K for C⊂( R + ) 2 a triangle T a,b with vertices (0,0),(b,0),(0,a) . Finally, we show how the definition of δ C (K) can be extended to include many nonconvex bodies C⊂ R d for d− circled sets K⊂ C d , and we prove an integral formula for δ C (K) which we use to compute a formula for the C− transfinite diameter of the Euclidean unit ball B⊂ C 2 .

It is well-known that the Carathéodory metric is a natural generalization of the Poincaré metric, namely, the hyperbolic metric of the unit disk. In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary proper subdomains of the complex plane and he proved that this metric coincides with the hyperbolic metric when the domains are simply connected. In this paper, we define a new metric which generalizes the Hurwitz metric in the sense of Carathéodory. Our main focus is to study its various basic properties in connection with the Hurwitz metric.

In the setting of quaternionic Heisenberg group H n−1 , we characterize the boundedness and compactness of commutator [b,C] for the Cauchy--Szegö operator C on the weighted Morrey space L p,κ w ( H n−1 ) with p∈(1,∞) , κ∈(0,1) and w∈ A p ( H n−1 ). More precisely, we prove that [b,C] is bounded on L p,κ w ( H n−1 ) if and only if b∈BMO( H n−1 ) . And [b,C] is compact on L p,κ w ( H n−1 ) if and only if b∈VMO( H n−1 ) .

In this study, a subclass of an univalent function with negative coefficients which is defined by a new general Linear operator have been introduced. The sharp results for coefficients estimators, distortion and closure bounds, Hadamard product, and Neighborhood, and this paper deals with the utilizing of many of the results for classical hypergeometric function, where there can be generalized to m-hypergeometric functions. A subclasses of univalent functions are presented, and it has involving operator which generalizes many well-known. Denote A the class of functions f and we have other results have been studied.

Let ϕ be a normalized convex function defined on open unit disk D . For a unified class of normalized analytic functions which satisfy the second order differential subordination f ′ (z)+αz f ′′ (z)≺ϕ(z) for all z∈D , we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.

In this article, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows, generalizing a result of Ransford-White for the spectral unit ball, that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider 2 -point and 3 -point Pick-Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by Bharali and Chandel related to these problems.

The purpose of the present paper is to find the necessary and sufficient conditions for the subclasses of analytic functions associated with Pascal distribution to be in subclasses of spiral-like univalent functions and inclusion relations for such subclasses in the open unit disk D. Further, we consider the properties of integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is (C,α) -summable to the function in the norm of the space, provided that α>1/2 . We further show that the constant 1/2 is sharp, in marked contrast with the classical case of the disk algebra.

We give a characterization of Nakano positivity of Riemannian flat vector bundles over bounded domains D⊂ R n in terms of solvability of the d equation with certain good L 2 estimate condition. As an application, we give an alternative proof of the matrix-valued Prekopa's theorem that is originally proved by Raufi. Our methods are inspired by the recent works of Deng-Ning-Wang-Zhou on characterization of Nakano positivity of Hermitian holomorphic vector bundles and positivity of direct image sheaves associated to holomorphic fibrations.

We solve Chui's conjecture on the simplest fractions (i.e., sums of Cauchy kernels with unit coefficients) in weighted (Hilbert) Bergman spaces. Namely, for a wide class of weights, we prove that for every N , the simplest fractions with N poles on the unit circle have minimal norm if and only if the poles are equispaced on the circle. We find sharp asymptotics of these norms. Furthermore, we describe the closure of the simplest fractions in weighted Bergman spaces, using an L 2 version of Thompson's theorem on dominated approximation by simplest fractions.