On Geometric Aspects of Quaternionic and Octonionic Slice Regular Functions

The aim of this paper is twofold. On the one hand, we enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the octonionic space. As applications, we obtain two Landau–Toeplitz type theorems for slice regular functions with respect to regular diameter and slice diameter, respectively, together with a Cauchy type estimate. Along with these results, we introduce some new and useful ideas, which also allow us to prove the minimum principle and one version of the open mapping theorem. On the other hand, we adopt a completely new approach to strengthen a version of boundary Schwarz lemma first proved in Ren and Wang (Trans Am Math Soc 369:861–885, 2017) for quaternionic slice regular functions. Our quaternionic boundary Schwarz lemma with optimal estimate improves considerably a well-known Osserman type estimate and provides additionally all the extremal functions.     

Analytic Operator Functions with Selfadjoint Coefficients

In this note, results of H. K. Wimmer [2] are extended to functions with values in the space of bounded linear operators in the Hilbert space \cal{H}$ , which are analytic on an annulus and have selfadjoint coefficients. The eigenvalues on the boundary of the spectrum of the functions considered in this note, corresponding to a fixed eigenvector possess a certain symmetry which is well known from the periphal spectrum of entrywise nonnegative matrices, due to the Perron-Frobenius theorem. Similar results hold for polynomials of type (1) with entrywise nonnegative matrix coefficients A_j, see [1]. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Note on Laurent Series of the Octonionic Analytic Functions

In this note we obtain mainly the standard Laurent series for the octonionic analytic functions defined in an annulus of 8, which improves the earlier result of Li, Zhao and Peng [Adv. Appl. Clifford Alg. 11 (S2) (2001), 205-217].     

On Analytic Functions with Divergent Series of Taylor Coefficients

We consider a bounded, circular, strictly convex domain \(\Omega \) with \(C^{2}\) boundary. We show that there exists a holomorphic function \(f_{1}\), continuous to the boundary such that every slice function has a series of Taylor coefficients divergent with every power \(p\in [0,2)\). We also construct inner function \(f_{2}\) which every slice is also inner and has series of Taylor coefficients with the same property. Next we generalize it to obtain \(f_{3}\in {\mathbb {O}}(\Omega )\) with given modulus a.e. on all slices and Taylor series as above.     

可变幅角复系数的解析函数族

研究了在单位开圆盘内单叶解析且规范化的复系数函数族gφ1,φ2,φ3,φ4（m1,m2,m3,m4;λ）的一些性质,给出了其子族gφ1,φ2,φ3,φ4（m1,m2,m3,m4;λ）在内闭一致收敛拓扑下的极值点和支撑点,并讨论解决了gφ1,φ2,φ3,φ4（m1,m2,m3,m4;λ）与凸函数相关的一些半径问题,推广了近来的一些研究结果．     

Analytic Functions with Negative Coefficients and Differential and Integral Operators

In this paper integral properties of some analytic functions with negative coefficients from certain classes of functions defined using Ruscheweyh differential operator are studied. The obtained results are sharp and they are improvement of some known results.     

Improved Bounds for Taylor Coefficients of Analytic Functions

The practical computation of verified bounds for Taylor coefficients of analytic functions is considered. Using interval arithmetic, the bounds are constructed from Cauchy's estimate and from some of its modifications. By employing the mean value form for intermediate function evaluations, the accuracy of the bounds is improved by several powers of ten, compared to earlier results.     

具有缺项系数的几类解析函数族的性质

S*表示所有在单位圆盘 D 内解析且满足条件 f(0)=f′ (0)-1=0的星形函数族, K 表示所有在 D内解析且满足条件 f(0)=f′ (0)-1=0 的凸函数族, P 表示所有在 D 内解析且满足条件p(0)=1, Rep(z)>0 的函数族. 设P_n={p(z): p(z)=1+a_nzⁿ+a_n+1zⁿ⁺¹+…∈ P}, S*_n={f (z): f(z)=z+a_nzⁿ+a_n+1zⁿ⁺¹+…∈ S*}, K_n={f (z): f (z)=z+a_nzⁿ+a_n+1zⁿ⁺¹+…∈ K}. L_Sn*={g(z)=ln f(z)/z, f ∈ S_n*}, 其中对数函数取使得ln1=0的那个单值解析分支. 该文研究了函数族S_n^*, K_n和L_Sn*的性质, 找出了解析函数族L_Sn*的极值点与支撑点,并对S*_n与K_n的极值点和支撑点作了一些探讨.     

Orthogonal Invariance of the Dirac Operator and the Critical Index of Subharmonicity for Octonionic Analytic Functions

We show that the Dirac operator ${D = \sum_{0}^{7} e_k \frac{\partial}{\partial _{{x}_k}}}$ is orthogonal invariant. As an application, we give a new proof of the theorem in [7].     

On the Multiplicity of Zeroes of Polynomials with Quaternionic Coefficients

Regular polynomials with quaternionic coefficients admit only isolated zeroes and spherical zeroes. In this paper we prove a factorization theorem for such polynomials. Specifically, we show that every regular polynomial can be written as a product of degree one binomials and special second degree polynomials with real coefficients. The degree one binomials are determined (but not uniquely) by the knowledge of the isolated zeroes of the original polynomial, while the second degree factors are uniquely determined by the spherical zeroes. We also show that the number of zeroes of a polynomial, counted with their multiplicity as defined in this paper, equals the degree of the polynomial. While some of these results are known in the general setting of an arbitrary division ring, our proofs are based on the theory of regular functions of a quaternionic variable, and as such they are elementary in nature and offer explicit constructions in the quaternionic setting. Partially supported by G.N.S.A.G.A.of the I.N.D.A.M. and by M.I.U.R.. Lecture held by G. Gentili in the Seminario Matematico e Fisico on February 12, 2007. Received: August 2008     

Complex Interpolation of Nonlinear Operators and Lebedev-Milin Inequalities for Taylor Coefficients of Composed Analytic Functions

A generalization of a theorem of Bergh concerning complex interpolation of nonlinear operators is applied to prove Lebedev-Milin type estimates for Taylor coefficients of composed analytic functions.     

Functions Operating on Positive Semidefinite Quaternionic Matrices

 We study functions on the quaternionic unit ball which operate on positive semidefinite matrices in the sense that is positive semidefinite whenever is a positive semidefinite square matrix with entries . (Received 21 September 2000; in revised form 8 March 2001)     

Functions Operating on Positive Semidefinite Quaternionic Matrices

 We study functions on the quaternionic unit ball which operate on positive semidefinite matrices in the sense that is positive semidefinite whenever is a positive semidefinite square matrix with entries .     

有负系数的半纯凸函数

本文论述了一些有关设施农业建筑设计参数、环境工程设备的调节控制指标的选定。提出农业气象学是农业建筑和环境工程学的重要基础学科之一。     

具有缺项和负系数的多元函数

Let P_k(p, A, B) be the class of functions (?) k ≥2 analytic in the unit disc E={z:|z|<1} and satisfying the condition |(zf′(z)/f(z)-p)/(Ap-Bzf(z)/f(z))|<1. for z∈E and -1≤Bk(p, A, B) under the assumption - 1 ≤B< 0 .     

Analytic Functions Related with the Hyperbola

The author considers a new class SHm λμ(α) of normalized analytic functions defined by a diferential operator. Several basic properties and characteristics of the functions belonging to the class SHm λμ(α) are investigated. These include integral representations, coefcient bounds, the Fekete-Szego¨ problem, class-preserving operators and T δ-neighborhoods.     

Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

We consider Dirichlet series z_g,a(s)=?_n=1^￥ g(na) e^{-l_n s}{\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ _n  = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ?_n=1^￥ g(na) zⁿ{\sum_{n=1}^{\infty} g(n\alpha) z^n}. We prove that a Dirichlet series z_g,a(s) = ?_n=1^￥ g(n a)/n^s{\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s} has an abscissa of convergence σ ₀ = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ ₀ satisfies σ ₀ ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ _g,α (s) has an analytic continuation to the entire complex plane.     

关于具有负系数和缺系数的单叶函数

Let P_k(A,B) be the class of functions(?), regu-lar in the unit disc U={z:|z|<1} and satisfying |[f′(z)-1]/[A-Bf′(z)]|<1 for z∈U, where -1≤Bk(A,B) under the assumption -1≤B≤0. We also obtain class preserving integral operators of the form.     

On the Derivatives of Quaternionic Functions Along Two-Dimensional Planes

In a previous paper we introduced the concept of a two-dimensional directional derivative of a quaternionic function along a two-dimensional plane. In this paper we provide a deeper analysis of its properties, as well as of its relations with hyperholomorphic functions, with holomorphic maps of two complex variables and with Cullen-regular functions. Received: October, 2007. Accepted: February, 2008.