Radius Problems for Starlike Functions Associated with the Tan Hyperbolic Function

The aim of this particular article is at studying a holomorphic function f defined on the open-unit disc D = z ∈ ℂ : z < 1 for which the below subordination relation holds z f ′ z / f z ≺ q 0 z = 1 + tan h z . The class of such functions is denoted by S tan h ∗ . The radius constants of such functions are estimated to conform to the classes of starlike and convex functions of order β and Janowski starlike functions, as well as the classes of starlike functions associated with some familiar functions.

An improved characterisation of regular generalised functions of white noise and an application to singular SPDEs

A characterisation of the spaces $${\mathcal {G}}_K$$ G K and $${\mathcal {G}}_K'$$ G K ′ introduced in Grothaus et al. (Methods Funct Anal Topol 3(2):46–64, 1997) and Potthoff and Timpel (Potential Anal 4(6):637–654, 1995) is given. A first characterisation of these spaces provided in Grothaus et al. (Methods Funct Anal Topol 3(2):46–64, 1997) uses the concepts of holomorphy on infinite dimensional spaces. We, instead, give a characterisation in terms of U-functionals, i.e., classic holomorphic function on the one dimensional field of complex numbers. We apply our new characterisation to derive new results concerning a stochastic transport equation and the stochastic heat equation with multiplicative noise.

On properties of h-differentiable functions

Research in the theory of functions of an h-complex variable is of interest in connection with existing applications in non-Euclidean geometry, theoretical mechanics, etc. This article is devoted to the study of the properties of h-differentiable functions. Criteria for h-differentiability and h-holomorphy are found, formulated and proved a theorem on finite increments for an h-holomorphic function. Sufficient conditions for h-analyticity are given, formulated and proved a uniqueness theorem for h-analytic functions.

A generalization of Milnor’s formula

The Milnor number $$\mu _f$$ μ f of a holomorphic function $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ f : ( C n , 0 ) → ( C , 0 ) with an isolated singularity has several different characterizations as, for example: 1) the number of critical points in a morsification of f, 2) the middle Betti number of its Milnor fiber $$M_f$$ M f , 3) the degree of the differential $${\text {d}}f$$ d f at the origin, and 4) the length of an analytic algebra due to Milnor’s formula $$\mu _f = \dim _{\mathbb {C}}{\mathcal {O}}_n/{\text {Jac}}(f)$$ μ f = dim C O n / Jac ( f ) . Let $$(X,0) \subset ({\mathbb {C}}^n,0)$$ ( X , 0 ) ⊂ ( C n , 0 ) be an arbitrarily singular reduced analytic space, endowed with its canonical Whitney stratification and let $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ f : ( C n , 0 ) → ( C , 0 ) be a holomorphic function whose restriction f|(X, 0) has an isolated singularity in the stratified sense. For each stratum $${\mathscr {S}}_\alpha $$ S α let $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) be the number of critical points on $${\mathscr {S}}_\alpha $$ S α in a morsification of f|(X, 0). We show that the numbers $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) generalize the classical Milnor number in all of the four characterizations above. To this end, we describe a homology decomposition of the Milnor fiber $$M_{f|(X,0)}$$ M f | ( X , 0 ) in terms of the $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) and introduce a new homological index which computes these numbers directly as a holomorphic Euler characteristic. We furthermore give an algorithm for this computation when the closure of the stratum is a hypersurface.

On some properties of p-holomorphic and p-analytic function

In this article the relationship between the conditions of p-differentiability, p-holomorphycity, and the existence of the derivative of a function of a p-complex variable is considered. The general form of a p-holomorphic function is found. The sufficient conditions for p-analyticity and local invertibility are obtained. The open mapping theorem and the principle of maximum of the norm for a p-holomorphic function and the uniqueness theorem are proved.

Radon Inversion Problem for Holomorphic Functions on Circular, Strictly Convex Domains

In this paper we study the so-called Radon inversion problem in bounded, circular, strictly convex domains with $${\mathcal {C}}^2$$ C 2 boundary. We show that given $$p>0$$ p > 0 and a strictly positive, continuous function $$\Phi $$ Φ on $$\partial \Omega $$ ∂ Ω , by use of homogeneous polynomials it is possible to construct a holomorphic function $$f \in {\mathcal {O}}(\Omega )$$ f ∈ O ( Ω ) such that $$\displaystyle \smallint _0^1 |f(zt)|^pdt = \Phi (z)$$ ∫ 0 1 | f ( z t ) | p d t = Φ ( z ) for all $$z \in \partial \Omega $$ z ∈ ∂ Ω . In our approach we make use of so-called lacunary K-summing polynomials (see definition below) that allow us to construct solutions with in some sense extremal properties.

Majorization Results for Certain Subfamilies of Analytic Functions

Let h 1 z and h 2 z be two nonvanishing holomorphic functions in the open unit disc with h 1 0 = h 2 0 = 1 . For some holomorphic function q z , we consider the class consisting of normalized holomorphic functions f whose ratios f z / z q z and q z are subordinate to h 1 z and h 2 z , respectively. The majorization results are obtained for this class when h 1 z is chosen either h 1 z = cos z or h 1 z = 1 + sin z or h 1 z = 1 + z and h 2 z = 1 + sin z .