Bifurcation sets of definable functions in o-minimal structures

In this work we answer a question stated by Loi and Zaharia concerning trivialization of definable functions off the bifurcation set: we prove that definable functions are trivial off the bifurcation set, and the trivialization can be chosen definable.

     

Fiberwise properties of definable sets and functions in o-minimal structures

Summary We examine how in any o-minimal expansion of a dense linear order, fiberwise open implies pecewise open for sets definable with parameters, and fiberwise continuous implies piecewise continuous for functions definable with parameters.     

Tame properties of sets and functions definable in weakly o-minimal structures

Let ${{\mathcal{M}}=(M, <, \ldots )}$ be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in ${{\mathcal{M}}}$ which hold in o-minimal structures, are examined. One of them is the intermediate value property, say IVP. It is shown that strongly continuous definable functions in ${{\mathcal{M}}}$ satisfy an extended version of IVP. After introducing a weak version of definable connectedness in ${{\mathcal{M}}}$ , we prove that strong cells in ${{\mathcal{M}}}$ are weakly definably connected, so every set definable in ${{\mathcal{M}}}$ is a finite union of its weakly definably connected components, provided that ${{\mathcal{M}}}$ has the strong cell decomposition property. Then, we consider a local continuity property for definable functions in ${{\mathcal{M}}}$ and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph (NDG) which was examined for definable functions in (Dolich et al. in Trans. Am. Math. Soc. 362:1371–1411, 2010) and related to uniform finiteness, definable completeness, and others. We show that every weakly o-minimal structure ${{\mathcal{M}}}$ having cell decomposition, satisfies NDG, i.e. every definable function in ${{\mathcal{M}}}$ has no dense graph.     

单连通区域上解析函数的插值问题

本文利用单位圆盘上Hardy空间插值问题的已知结论,用较初等的方法,对边界至少含有两个不同点的任意单连通区域,给出插值问题有解的充分必要条件。     

On classes of analytic functions defined by convolution with incomplete beta functions

Carlson and Shaffer [SIAM J. Math. Anal. 15 (1984) 737-745] defined a convolution operator L(a,c) on the class A of analytic functions involving an incomplete beta function ?(a,c;z) as L(a,c)f=?(a,c)?f. We use this operator to introduce certain classes of analytic functions in the unit disk and study their properties including some inclusion results, coefficient and radius problems. It is shown that these classes are closed under convolution with convex functions.     

Certain subclasses of multivalent analytic functions defined by multiplier transforms

By making use of the principle of subordination between analytic functions and a family of multiplier transforms, we introduce and investigate some new subclasses of multivalent analytic functions. Such results as inclusion relationships, subordination and superordination properties, integral-preserving properties, argument estimates and convolution properties are proved.     

The lack of definable witnesses and provably recursive functions in intuitionistic set theories

Let ZFI_R(ZFI_C) be intuitionistic ZF set theory formulated with Replacement (resp. Collection). It is known that if ZFI_R proves a sentence ∃xA(x), then there is a formula C(z) so that ZFI_R proves ∃!zC(z) and ∃x(C(x) ∧ A(x)), the existence property. It is shown that ZFI_C does not have the existence property, and thus ZFI_R ⫋ ZFI_C. This remains true even if one adds Dependent Choice and all true Σ₁ sentence of ZF. It is known that ZF and ZFI_c have the same provably recursive functions. It is also shown that this is not true for ZFI_C and ZFI_R.     

Univalence and starlikeness of certain transforms defined by convolution of analytic functions

Let U(λ) denote the class of all analytic functions f in the unit disk Δ of the form f(z)=z+a₂z²+? satisfying the condition

     

On certain class of multivalent analytic functions defined by differential subordination

By making use of the principle of subordination, we introduce a certain class of multivalent analytic functions. Such results as subordination and superordination properties, distortion theorems and inequality properties are proved. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.     

Subordination results for certain class of analytic functions defined by convolution

In this paper, we drive several interesting subordination results of a certain class of analytic functions defined by convolution.     

On the analytic continuation of functions defined by regrouped power series

We consider functions defined by regrouped power series \(f(z) = \sum\nolimits_{n = 0}^\infty z ^{\lambda _{n_{P_{k_n } } } } (z)\) in the disk |z|<1 and also in some domain D outside of this disk. We obtain conditions under whichf(z) is analytically continuable outside of the disk |z|<1, the analytic continuation being effected with the help of the given series. We also consider the analytic continuability of functionsf(z, w).     

On certain classes of analytic functions defined by Noor integral operator

Integral operator, introduced by Noor, is defined by using convolution. Let f_n(z)=z/(1−z)ⁿ⁺¹, n∈N₀, and let f be analytic in the unit disc E. Then I_nf=f⁽⁻¹⁾_n★f, where f_n★f⁽⁻¹⁾_n=z/(1−z). Using this operator, certain classes of analytic functions, related with the classes of functions with bounded boundary rotation and bounded boundary radius rotation, are defined and studied in detail. Some basic properties, rate of growth of coefficients, and a radius problem are investigated. It is shown that these classes are closed under convolution with convex functions. Most of the results are best possible in some sense.     

Majorization for certain classes of analytic functions defined by fractional derivatives

Here we investigate a majorization problem involving starlike function of complex order belonging to a certain class defined by means of fractional derivatives. Relevant connections of the main results obtained in this paper with those given by earlier workers on the subject are also pointed out.     

Coefficient estimates for a certain class of analytic and bi-univalent functions defined by fractional derivative

We introduce and investigate a subclass of analytic and bi-univalent functions defined by a fractional derivative operator in the open unit disk. Using the Faber polynomial expansions, we obtain upper bounds for the coefficients of functions belonging to this class.     

Subordination properties for a certain class of analytic functions defined by the Salagean operator

In this paper we derive several subordination results for a certain class of analytic functions defined by the Salagean operator.     

On the singularities of analytic functions defined by L-Dirichletian elements

In this note, using the Dzrbasjan-Mandelbrojt inequality, we locate the singularities of analytic functions defined by L-Dirichletian elements. Though Dzrbasjan remarks that his inequality can be used to locate the singularities of functions defined by the Dirichlet-Taylor series, it is the convergence theory we established earlier ¦M. Blambert and M. Berland, C. R. Acad. Sci. Paris280 (1975), 263–266; M. Blambert and R. Parvatham, Ann. Inst. Fourier29 (1979), 239–262¦ that leads us exactly to determine the natural boundaries of such functions, whereas Dzrbasjan is never concerned with the convergence of such a series, as he himself states.     

Arithmetic properties of values of analytic functions connected by algebraic equations over fields of rational functions

In this paper theorems are proved about the arithmetic character of the values at algebraic points of a collection of G-functions which constitute a solution of a system of linear differential equations with coefficients from C(z) connected by algebraic equations over C(z). In addition, the theorems on E-functions proved by A. B. Shidlovskii in 1962 are supplemented.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 83–94, July, 1973.In conclusion, I express thanks to A. B. Shidlovskii for attention to and help with this work.