On locally biholomorphic finitely valent mappings from multi-connected domains onto the open disc

In this paper we offer new results of research presented in the referred papers of J.E. Fornaess and E.L. Stout, E. Ligocka and the authors, and concerning the existence of m-valent locally biholomorphic mappings from product domains of Cn onto n-dimensional complex manifolds. In particular, we confirm an own conjecture about the estimation of the valentness m of locally biholomorphic mappings from multi-connected domains onto the open unit disc.     

Finitely valent locally biholomorphic mappings of bounded domains

Considering countably connected domains of the plane, we solve the problem of existence of finitely valent locally biholomorphic mappings of a bounded domain onto a disk. We obtain sufficient conditions for the existence of the corresponding mappings in higher dimensions.     

q-subharmonicity and q-convex domains in ℂ n

In this paper we study q-subharmonic and q-plurisubharmonic functions in ? ⁿ . Next as an application, we give the notion of q-convex domains in ? ⁿ which is an extension of weakly q-convex domains introduced and investigated in [10]. In the end of the paper we show that the q-convexity is the local property and give some examples about q-convex domains.     

The Berezin transform and Laplace-Beltrami operator

We study the Berezin transform of bounded operators on the Bergman space on a bounded symmetric domain Ω in Cn. The invariance of range of the Berezin transform with respect to G=Aut(Ω), the automorphism group of biholomorphic maps on Ω, is derived based on the general framework on invariant symbolic calculi on symmetric domains established by Arazy and Upmeier. Moreover we show that as a smooth bounded function, the Berezin transform of any bounded operator is also bounded under the action of the algebra of invariant differential operators generated by the Laplace-Beltrami operator on the unit disk and even on the unit ball of higher dimensions.     

The existence of k-convex hypersurface with prescribed mean curvature

Using the strong maximum principle, we obtain a constant rank theorem for the k-convex solutions of semilinear elliptic partial differential equations. As an application we obtain an existence theorem of k-convex starshaped hypersurface with prescribed mean curvature in R ⁿ⁺¹.     

The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ? starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ? starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ? starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.     

A decomposition theorem for χ1-convex sets

A set S in $\text{R}$ is said to be χ-convex if and only if S does not contain a visually independent subset having cardinality χ. It is natural to ask when an χ-convex set may be expressed as a countable union of convex sets. Here it is proved that if S is a closed χ-convex set in the plane and $\text{R}$ has at most finitely many bounded components, then S is a countable union of convex sets. A parallel result holds in $\text{R}$ when S is a closed χ-convex set which contains all triangular regions whose relative boundaries are in S. However, the result fails for arbitrary χ-convex sets, even in the plane.     

Homotopy formulas and\bar \partial -equation on localq-convex domains in Stein manifolds-equation on localq-convex domains in Stein manifolds

The homotopy formulas of (r, s) differential forms and the solution of $\bar \partial $ -equation of type (r, s) on localq-convex domains in Stein manifolds are obtained. The homotopy formulas on localq-convex domains have important applications in uniform estimates of $\bar \partial $ -equation and holomorphic extension of CR-manifolds.     

Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces

In this paper, we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball B in a complex Hilbert space X is a biholomorphic convex mapping, which improves some results of Hamada and Kohr and solves the problem which is posed by Graham and Kohr. From this, we derive some sufficient conditions for biholomorphic convex mapping. We also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on B in Hilbert spaces. Moreover, we give some examples of biholomorphic convex mappings on B in Hilbert spaces.     

Uniform and refinable grids on elliptic domains and on some surfaces of revolution

We construct a bijection from R² to R², which maps rectangles centered at the origin O onto ellipses centered at O, and preserves area. This bijection allows us to construct uniform and refinable grids on elliptic domains. Then, we combine a particular case of this bijection (i.e. that bijection that maps squares into circles) with another area preserving projection from R² to a surface of revolution around Oz. This yields uniform and refinable grids on this surface of revolution. The lines of these grids are situated in horizontal planes, if they are images of squares centered at O. We consider the particular case of the hemisphere and show how the northern hemisphere of the Earth is projected onto a square. Thus, our equiareal maps can be useful for constructing geographical maps of one hemisphere of the Earth onto rectangles.     

Global Boundary Regularity for the {\overline\partial}-problem on Strictly q-convex and q-concave Domains

We prove the boundary global regularity of the ${\overline\partial}$ -operator on strictly q-convex and q-concave domains in K?hler manifolds. Applications to the solvability of the tangential Cauchy?CRiemann equations for smooth forms on boundaries of such domains are given.     

General decomposition theorems form-convex sets in the plane

A setS inR ^dis said to bem-convex,m≧2, if and only if for everym distinct points inS, at least one of the line segments determined by these points lies inS. Clearly any union ofm?1 convex sets ism-convex, yet the converse is false and has inspired some interesting mathematical questions: Under what conditions will anm-convex set be decomposable intom?1 convex sets? And for everym≧2, does there exist aσ(m) such that everym-convex set is a union ofσ(m) convex sets? Pathological examples convince the reader to restrict his attention to closed sets of dimension≦3, and this paper provides answers to the questions above for closed subsets of the plane. IfS is a closedm-convex set in the plane,m ≧ 2, the first question may be answered in one way by the following result: If there is some lineH supportingS at a pointp in the kernel ofS, thenS is a union ofm ? 1 convex sets. Using this result, it is possible to prove several decomposition theorems forS under varying conditions. Finally, an answer to the second question is given: Ifm≧3, thenS is a union of (m?1)³2 ^m?3 or fewer convex sets.     

Six Kinds of Roughly Convex Functions

This paper considers six kinds of roughly convex functions, namely: δ-convex, midpoint δ-convex, ρ-convex, γ-convex, lightly γ-convex, and midpoint γ-convex functions. The relations between these concepts are presented. It is pointed out that these roughly convex functions have two optimization properties: each r-local minimizer is a global minimizer, and if they assume their maximum on a bounded convex domain D (in a Hilbert space), then they do so at least at one r-extreme point of D, where r denotes the roughness degree of these functions. Furthermore, analytical properties are investigated, such as boundedness, continuity, and conservation properties.     

The Busemann theorem for complex p-convex bodies

The Busemann theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper, we prove a version of the Busemann theorem for complex p-convex bodies. Namely that the complex intersection body of an origin-symmetric complex p-convex body is γ-convex for certain γ. The result is the complex analogue of the work of Kim, Yaskin, and Zvavitch on (real) p-convex bodies. Furthermore, we show that the generalized radial qth mean body of a p-convex body is γ-convex for certain γ.     

Sufficient Conditions and Duality for a Class of Multiobjective Fractional Programming Problems with Higher-Order (F,α,ρ,d)-Convexity

In this paper, a class of multiobjective fractional programming problems (denoted by (MFP)) is considered. First, the concept of higher-order (F,α,ρ,d)-convexity of a function f:C→R with respect to the differentiable function φ:R ⁿ ×R ⁿ →R is introduced, where C is an open convex set in R ⁿ and α:C×C→R ₊?{0} is a positive value function. And an important property, which the ratio of higher-order (F,α,ρ,d)-convex functions is also higher-order (F,α ^′,ρ ^′,d ^′)-convex, is proved. Under the higher-order (F,α,ρ,d)-convexity assumptions, an alternative theorem is also given. Then, some sufficient conditions characterizing properly (or weakly) efficient solutions of (MFP) are obtained from the above property and alternative theorem. Finally, a class of dual problems is formulated and appropriate duality theorems are proved.     

A better decomposition theorem for simply connected m-convex sets

A set S in R^d is said to be m-convex, m ? 2, if and only if for every m points in S, at least one of the line segments determined by these points lies in S. For S a closed m-convex set in R², various decomposition theorems have been obtained to express S as a finite union of convex sets. However, the previous bounds may be lowered further, and we have the following result:In case S is simply connected, then S is a union of σ(m) or fewer convex sets, where $\text{σ(m) = [(m ?}\text{N}\text{)(m ?}\text{3}\text{2}\text{) +}\text{3}\text{2}\text{]}$.Moreover, this result induces an improved decomposition in the general case as well.     

On the space of B-convex compacta

We investigate topology of the space of B-convex compacta of finite-dimensional Banach space (the notion of B-convexity space was introduced by M. Lassak). An answer to the question of M. van de Vel about a characterization of continuity of the closed B-convex hull is given. We prove that the space of B-convex compacta is a Q-manifold iff the map of the closed B-convex hull is continuous.     

Hölder andL p -estimates for the\bar \partial -equation on non-smooth strictlyq-convex domainson non-smooth strictlyq-convex domains

A solution operator for the \(\bar \partial \) -equation on strictlyq-convex domains with nonsmooth boundary is constructed. It is proved that the solution satisfies optimal 1/2-Hölder andL ^p estimates.     

Positively regular homeomorphisms of Euclidean spaces

A homeomorphism of Rⁿ onto itself is called positively regular (or EC⁺) iff its family of non-negative iterates is pointwise equicontinuous. For EC⁺ homeomorphism of Rⁿ such that some point of Rⁿ has bounded positive semi-orbit, the nucleus M is defined, and the following theorems are proved.Theorem 1. If such a homeomorphism h:Rⁿ→Rⁿ has compact nucleus M, then M is a fully invariant compact AR. Further, for n≠4,5,h:Rⁿ/M→Rⁿ/M is conjugate to a contraction on Rⁿ.Theorem 2. In Rⁿ,n≠4,5,M compact iff there existsa disk D such that h(D)?IntD.Theorem 3. In R², either M is a disk and h|M is a rotation, or h|M is periodic. The relationship between M and the irregular set of ? is also studied.     

Nonlinear programming with E-preinvex and local E-preinvex functions

In this paper, we extend the class of E-convex sets, E-convex and E-quasiconvex functions introduced by [Youness, E.A., 1999. E-convex sets, E-convex functions and E-convex programming. Journal of Optimization Theory and Applications 102, 439–450], respectively by [Syau, Yu-Ru, Lee, E. Stanley, 2005. Some properties of E-convex functions. Applied Mathematics Letters 18, 1074–1080] to E-invex set, E-preinvex, E-prequasiinvex and corresponding local concepts. Some properties of these classes are studied. As an application of our results, we consider the nonlinear programming problem for which, we establish that, under mild conditions, a local minimum is a global minimum.