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In Geometric function theory, the Ma-Minda class of starlike functions has a unique place as it unifies various subclasses of starlike functions. There has been an vivid interplay between special functions and their geometric properties, like starlikeness. In this article, we establish certain special function's radius of Ma-Minda starlikness. As an application, we obtain conditions on parameters for these special functions to be in the Ma-Minda class. Further, we focus on certain convolution properties for the Ma-Minda class that are not done so far, and study their applications in radius problem. Finally, we prove a variational problem of Goluzin, namely, the region of variability for the Ma-Minda class. Our results simplify and generalize the already-known ones. [ABSTRACT FROM AUTHOR]

In this article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V(k) with every Hardy space Hp is closed in that Hp and associate singular continuous measures to elements of V(k). There have been no examples in the literature of functions in V(k) with zeros in the unit disk other than the one at the origin. We close this gap in the literature. We derive a representation of the measure associated to a function in V(k) for functions whose derivatives are rational, or algebraic, or transcendental functions in the unit disk.Finally, we consider the notion of regulated domains, introduced by Pommerenke and show that there are regulated domains whose boundary is not of bounded boundary rotation.

Let 𝓐 denote the family of analytic functions f with f(0) = f′(0) – 1 = 0, in the open unit disk Δ. We consider a class S c s ∗ (α) := f ∈ A : z f ′ (z) f (z) − 1 ≺ z 1 + α − 1 z − α z 2 , z ∈ Δ , $$\begin{array}{} \displaystyle \mathcal{S}^{\ast}_{cs}(\alpha):=\left\{f\in\mathcal{A} : \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1+\left(\alpha-1\right) z-\alpha z^2}, \,\, z\in \Delta\right\}, \end{array}$$ where 0 ≤ α ≤ 1/2, and ≺ is the subordination relation. The methods and techniques of geometric function theory are used to get characteristics of the functions in this class. Further, the sharp inequality for the logarithmic coefficients γn of f ∈ S c s ∗ $\begin{array}{} \mathcal{S}^{\ast}_{cs} \end{array}$ (α): ∑ n = 1 ∞ γ n 2 ≤ 1 4 1 + α 2 π 2 6 − 2 L i 2 − α + L i 2 α 2 , $$\begin{array}{} \displaystyle \sum_{n=1}^{\infty}\left|\gamma_n\right|^2 \leq \frac{1}{4\left(1+\alpha\right)^2}\left(\frac{\pi^2}{6}-2 \mathrm{Li}_2\left(-\alpha\right)+ \mathrm{Li}_2\left(\alpha^2\right)\right), \end{array}$$ where Li2 denotes the dilogarithm function are investigated. [ABSTRACT FROM AUTHOR]

We aim to present a new linear integral operator and a certain subclass of regular functions with bounded turning property in correlation with the extended generalised Mittag-Leffler functions. Several results concerning an application of differential subordination are also investigated. [ABSTRACT FROM AUTHOR]

The aim of this article is to give an overview of the nature of current lexicographic user guidance devices and to suggest a possible classification or taxonomy to serve as a guideline for future compilations of such tools to assist users in communicative and cognitive situations, especially in text production, text reception and computer assisted vocabulary learning. The higher levels of the taxonomy are broadly based on the Function Theory of Lexicography. The lower level of tools that support the communicative function describes the nature of the interaction of the user with the tool - interactive, automated linking/checking and automated translation. The lower level of the tools that describe the cognitive function can be further expanded, based on the nature of the cognition e.g. into self-study of corpus-based analyses and computer-aided learning. We expect this work to provide a broader basis for the discussion of this dispersed set of tools, and for a more principled design and development of new ones. [ABSTRACT FROM AUTHOR]

Though a vast amount of dictionary analyses have been produced over the years, hardly any of these have mentioned the operative function, which has been overlooked in most lexicographical literature. With short analyses of 12 existing dictionaries ranging from the 18th century to the 21st century, this article shows that many dictionaries have indeed been produced to satisfy operative needs. Based on this result, it is clear that the operative function deserves a place in lexicographical theory. An interesting finding that came out of these analyses was that especially dictionaries from the 18th to the early 20th centuries (the old dictionaries) were written to accommodate several types of information needs that their users would come across in the real world, including operative needs, whereas the focus of most contemporary dictionaries is to satisfy linguistic information needs. This is an interesting change in focus, which this article criticises. Based on the abovementioned analyses, a number of questions are raised to guide future research into the operative function. [ABSTRACT FROM AUTHOR]

A trigonometric sine series with special properties is constructed. One of the properties of this series shows that Bary’s conjecture on subsequences of partial sums of a sine series is not valid. [ABSTRACT FROM AUTHOR]

We consider the evolution of open planar curves by the steepest descent flow of a geometric functional, with different boundary conditions.We prove that, if any set of stationary solutions with fixed energy is finite, then a solution of the flow converges to a stationary solution as time goes to infinity. We also present a few applications of this result. [ABSTRACT FROM AUTHOR]

We use the minimizing movement theory to study the gradient flow associated to a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, we can define a Willmore energy on regular functions of L1(Rd). This functional is extended to every L1 function by taking its lower semicontinuous envelope. We study the flow generated by this relaxed energy for radially non-increasing functions (functions with balls as superlevel sets). In the first part of the paper, we prove a coarea formula for the relaxed energy of such functions. Then, we show that the flow consists of an erosion of the initial data. The erosion speed is given by a first order ordinary equation. [ABSTRACT FROM AUTHOR]

We study projective dimension and graded length of structural modules in parabolic-singular blocks of the BGG category O. Some of these are calculated explicitly, others are expressed in terms of two functions. We also obtain several partial results and estimates for these two functions and relate them to monotonicity properties for quasi-hereditary algebras. The results are then applied to study blocks of O in the context of Guichardet categories, in particular, we show that blocks of O are not always weakly Guichardet. [ABSTRACT FROM AUTHOR]

It is a classical result in complex analysis that the class of functions that arise as the Cauchy transform of probability measures may be characterized entirely in terms of their analytic and asymptotic properties. Such transforms are a main object of study in non-commutative probability theory as the function theory encodes information on the probability measures and the various convolution operations. In extending this theory to operator-valued free probability theory, the analogue of the Cauchy transform is a non-commutative function with domain equal to the non-commutative upper-half plane. In this paper, we prove an analogous characterization of the Cauchy transforms, again, entirely in terms of their analytic and asymptotic behavior. We further characterize those functions which arise as the Voiculescu transform of -infinitely divisible operator-valued distributions. As consequences of these results, we provide a characterization of infinite divisibility in terms of the domain of the relevant Voiculescu transform, provide a purely analytic definition of the semigroups of completely positive maps associated to infinitely divisible distributions and provide a Nevanlinna representation for non-commutative functions with the appropriate asymptotic behavior. [ABSTRACT FROM AUTHOR]

In this work we present a new geometric approach to some problems in differential subordination theory. In the paper some sufficient conditions for function to be starlike or univalent or to be in the class of Carathéodory functions are obtained. We also discuss the new results closely related to the generalized Briot-Bouquet differential subordination. [ABSTRACT FROM AUTHOR]

In this paper, the notion of kth-order weighted slant Hankel operators on the space L²(β), β = {βn}n ∈ Z being a sequence of positive numbers with β0 = 1, is introduced and some of its algebraic and spectral properties are also discussed. The study is further extended to its compression on the space H² (β). [ABSTRACT FROM AUTHOR]

In this paper we prove the stability result for the functional equation (1/ L)∑λ∈K F( x + λ y) = G( x) + H( y), x, y ∈ S, for functions F, G, H: S → X, where ( S,+) is an abelian semigroup, K is a finite subgroup of the automorphism group of S, L := | K|, ( X,∥·∥) is the Banach space over a field 핂 ∈ {ℝ,ℂ}. [ABSTRACT FROM AUTHOR]

We show by a new example how to obtain a priori estimates for the velocity of the classical potential flow, using distortion theory for conformal mappings. [ABSTRACT FROM AUTHOR]