Complex Analysis and Applications

PREFACE

CONTENTS

Dirichlet and Neumann boundary value problems are considered for the inhomo-geneous Cauchy-Riemann equation in a quarter plane. Solvability conditions and solutions are given in explicit form.

Some real matrix representations of 4-dimensional Clifford numbers are presented and some properties of 4-dimensional Clifford numbers are obtained. As an application, an explicit expression of the general solution of equation axb = d and ax = xb are obtained in terms of a and b in 4-dimensional Clifford algebra Cℓ_0,3. As a by-product, we obtain the necessary and sufficient conditions for two Clifford numbers to be similar.

In 1987, Cameron and Storvick introduced change of scale formulas for Wiener integrals of bounded functions in a Banach algebra on classical Wiener space. In 1994, Yoo and Skoug extended these results to abstract Wiener space, and recently Chang, Kim, Song and Yoo obtained a change of scale formula of various functions which need not be bounded or continuous. Let C₀(𝔹) denote the space of all abstract Wiener space-valued continuous functions on [0, T] which vanish at 0. In this paper, we establish a change of scale formula for Wiener integrals of functions on C₀(𝔹) which need not be bounded or continuous. Using our formula, we obtain the above change of scale formulas for Wiener integrals of bounded or unbounded functions on classical and abstract Wiener spaces, as corollaries.

In this paper, we reveal the properties of solutions for a class of periodic differential equations with dominant coefficients of orders more extensive than positive integers or infinity for the first time, and simultaneously discover that the dominant coefficients for this class of equations only need to possess “single-side property”.

Q_p spaces on the unit disc were introduced, and their basic properties established, in 1995 by Aulaskari, Xiao and Zhao. Later some of these results were extended also to the unit ball or even to strictly pseudoconvex domains in the complex n-space. We briefly review the theory of bounded symmetric domains, of which the disc and the ball are the simplest examples, and then discuss the Q_p spaces in this setting. It turns out that some new phenomena appear, most notably concerning the relationships of these spaces to the various kinds of Bloch spaces on symmetric domains.

No abstract received.

No abstract received.

In this paper, we obtain two perturbation results on the linear differential function

In this paper, we study a special holomorphic family (M, π, R) of closed Riemann surfaces of genus two over a fourth punctured torus R, which is a kind of a Kodaira surface and is constructed by Riera. We give two explicit defining equations for (M, π, R) by using elliptic functions, and determine all the holomorphic sections of (M, π, R). Proofs will appear elsewhere.

An α-asymptotically conformal fixed point for a quasiconformal map defined on a domain is defined for any 0 ≤ α ≤ 1. Furthermore, we prove that for α > 0, there is a normal form for this kind attracting or repelling or super-attracting fixed point under quasiconformal changes of coordinate which is also asymptotically conformal at this point. These results generalize König's Theorem and Böttcher's Theorem in classical complex analysis. The idea in proofs is new and uses holomorphic motion theory and provides a new understanding of the inside mechanism of these two famous theorems.

This present paper investigate the growth problem of solution of certain high-order linear differential equations and obtain precise estimates of hyper-order of their solutions with infinite order.

Let be a family of holomorphic functions in a domain D, let k be a positive integer, and let a(z), b(z)(≠ 0) and c(z)(≠ 0) be holomorphic functions in D. If, for each and a complex number d, all zeros of f – d have multiplicity at least k, L[f] = a whenever f = 0, f = c whenever L[f] = b, where L[f] denotes linear differential polynomial in f with coefficients holomorphic in D, then is normal in D. This result extends and improves some normality criterions due to Miranda, Chuang and other authors. Some examples are provided to show that our results is sharp.

Using the Cesàro means of Taylor extensions of holomorphic functions f on the unit disc, an equivalent condition for f to belong to the Bloch type spaces is given.

In this paper, we introduce the definition of weakly weighted-sharing which is between “CM” and “IM”. Using the notion of weakly weighted-sharing, we study the uniqueness problems on meromorphic function and its kth order derivative f^(k). As consequences, we are able to answer questions posed by Kit-wing Yu, which were also studied by L. P. Liu and Y. X. Gu. Our results sharpen the above results.

In this paper, we deal with a problem of uniqueness for meromorphic functions defined in the whole plane C under certain value/set-sharing conditions in a sector instead of the plane.

In this paper, we study the convergent properties of Laplace-Stieltjes transforms and obtain some completely different results.

We give a new atomic decomposition for the Hardy space on domains.

We develop elementary properties of locally compact morphisms from coordinated manifolds transversal to submanifolds of coordinated manifolds as extension of regular values and submersions. Transversal submanifolds are treated as a special case. Examples are given at the end.

Define the logarithmic (α, β)-Bloch space and characterize the logarithmic (α, β)-Bloch functions by their Taytor coefficients. The relationship between logarithmic (α, β)-Bloch space and Cesàro means is given.

During the Edo period, mathematics was practised in Japan almost independently of European influence. Seki Takakazu and his disciple Takebe Katahiro learned from the Suanxue Qimeng how to express one-variable polynomials with numerical coefficients on the counting board and developed this idea using the so-called side writing method, by which they could treat one-variable polynomials with polynomial coefficients. (See ⁴.) This mathematical tool also helped them to investigate various properties of the circle. For example, they were interested in representing arc length s in terms of a given diameter d and a sagitta c. Since, in modern notation, s/2 = d arcsin , one of their results can be described as follows. Let and g_n(t) = g₁(g_n−1(t)) for n = 1,2,3, ⋯. Then Takebe recognized

In this paper, we mainly discuss the uniqueness problem of k order derivatives of meromorphic functions that share four small functions, and consider Nevanlinna's Four Values Theorem when the derivatives of meromorphic functions share some small functions, which generalize and improve some related results of the author.

Let φ be an analytic function on the open unit disk in the complex plane. We study the boundedness of the following operators

In this paper, we investigate the growth and regular growth of functions of infinite order defined by Laplace-Stieltjes transformations and obtain some necessary and sufficient conditions.

In this paper, a new singular direction of meromorphic function namely T-direction is investigated. Let a meromorphic function takes the value of its small function, we prove the existence of the corresponding T-direction.

This paper studies the Nevanlinna type classes and operators such as multipliers, nontangential maximal function operator and area operator. These operators are useful in complex and harmonic analysis.

The main goal of this paper is to extend A.Connes's result on Fredholm index to whole Hilbert space. A Fredholm module will be constructed by using singular integral operators with the Cauchy kernel and the relationship between the Fredholm index of the singular integral operators and K-group was set up.

Suppose X is a metric measure space. As a generalization of the p-modulus and the p-capacities in X, the Ψ-modulus and the Ψ-capacities are defined by means of the Luxemburg norm in the Orlicz space L^Ψ(X) with a general Yang function Ψ, instead of the ordinary norm in L^p(X). The relations between the Ψ-modulus and Ψ-capacities are discussed and some related equalities are established.

In this note, we give a characterization of Bloch function and little Bloch function, which extends results of Aulaskari-Lappan, Minda, and Aulaskari-Wulan.

In this paper, we investigate the uniqueness theory of algebroid functions and extend some uniqueness theorems of meromorphic functions (H.X. Yi [14], R. Nevanlinna [15,16]) to algebroid functions.

Let Δ be the unit disk in the complex plane. In this paper, we give a simple proof that there exists a point [μ] in the universal Teichmüller space T(Δ) which contains infinitely many extremal Beltrami coefficients but admits no extremal Teichmüller Beltrami coefficients.

In this paper, we proved a result that if two meromorphic functions f(z) and g(z) share four small functions a_j (z) (j = 1,⋯, 4) in the sense of E_k)(a_j, f) = E_k)(a_j,g), (j = 1, ⋯, 4) (k ≥ 11), then f is a quasi-Möbius transformation of g; i.e., there exist four small functions α_i(i = 1, ⋯, 4) of f and g such that f = (α₁g + α₂)/(α₃g + α₄), where α₁α₄− α₂α₃ ≢ 0.

In this paper we discuss the operator of multiplication M_ϕ between the Bloch-type spaces on the unit disc. Some sufficient and necessary conditions are given for which M_ϕ is a compact operator from B^α to B^β.

Let f(z) be a non-constant meromorphic function in the complex plane satisfying and d be a nonzero constant. Then

It is well known that, for 1 ≤ p < ∞, the diagonal Besov space B_p of the open unit ball admits a norm or semi-norm ∥ ∥_p such that ∥f ◦ φ∥_p = ∥f∥_p for all f in B_p and all automorphisms φ of the unit ball. We show here that the same result holds when 0 < p < 1.