The aim of this article is to introduce the notions of weak tautness and hyperconvexity of a domain in a Banach space and to establish the relation between them. Moreover, some conditions under which a balanced domain in a Hilbert space and a Hartogs domain in a Banach space are hyperconvex are also given.

We introduce fractional -integro-differentiation for functions holomorphic in the upper half-plane. It gives us a tool to construct Cauchy-Bergman type kernels associated with the weights Some estimates of the kernels enable us to obtain reproducing integral formulas for Bergman spaces with general weights which may decrease to zero with arbitrary rate near the origin. Accordingly, such Bergman functions have arbitrary growth near the real axis.

The aim of this article is to give explicit representations for solutions of the Robin boundary value problem for the Cauchy-Riemann operator [image omitted]. In the homogeneous cases we investigate the Robin boundary condition in a more general form. Finally, we give solutions of the corresponding higher-order operators.

This article studies the problem of the uniqueness of meromorphic functions that weighted sharing three values which improve some results given by Yi [Theorem 4, Yi, H.X., 1995, Unicity theorems for meromorphic functions that share three values. Kodai Mathematical Journal, 18, 300-314] and Ueda [Ueda, H., 1983, Unicity theorems for meromorphic or entire functions II. Kodai Mathematical Journal, 6, 26-36] and other authors. An application of these new results, if f and g are two distinct nonconstant meromorphic functions sharing 0, 1 and CM, and a is a nonconstant rational function, then N2)(r,1/(g-a))= [image omitted] An example shows that the latter result is not true for some transcendental small functions of f and g.

Let X be a smooth and connected Riemann surface of genus g 0 and f: X P1, h: X P1 non-constant meromorphic functions on X. Fix an integer n 4 and assume the existence of n distinct points a1, , an P1 such that [image omitted] (set-theoretically) for every i. Here we prove that either f = h or [image omitted].

The form of the Schiffer differential equation puts severe restrictions on the class of functions that can occur as extremal functions for arbitrary coefficient-functionals of finite degree. Theorem 1 characterizes the algebraic extremal functions for coefficient-functionals of finite degree. Furthermore it is shown that the extremal function either is an algebraic function or it must possess a non-isolated singularity, or must have a transcendental branch-point. The results are closely related to the Malmquist-Yosida theorems. However Nevanlinna's Theory of Value Distribution is not the mainly used tool but the special form of the Schiffer differential equation and the multiplicity of certain values together with the Great Picard Theorem are exploited.

In this article we shall extend the nonlinear superordination [image omitted] to the superordination of the form [image omitted]

In this article we establish conditions under which canonical variables can be defined for a variational problem defined on a geometric (compact) surface. Also, we show the form the corresponding Euler-Lagrange equations assume once we rewrite them in terms of such canonical variables. Furthermore, we prove a version of Jacobi's theorem generalizing the univariate standard version of this theorem. The main results are applied to the conformal Gauss curvature functional.

We construct a map from the space of Jacobi-like forms [image omitted]() for a discrete subgroup [image omitted] to the space [image omitted] of sequences of meromorphic functions satisfying certain conditions determined by some linear ordinary differential operators and prove that the Hecke operator actions on [image omitted]() and on [image omitted] are compatible with respect to this map.

We give conditions on the functions and u on [image omitted] such that if u is given by the convolution of and u, then u is harmonic on [image omitted].