Abstract

The objective of this paper is to introduce certain new subclass of $p$-valent analytic functions in the open unit disc $E=\{z:|z|<1\}$ and obtain sharp upper bound for the second Hankel determinant of functions belonging to this class, using Toeplitz determinants.

Keywords: $p$-valent analytic function; function whose derivative has a positive real part; second Hankel determinant; positive real function; Toeplitz determinants

MSC: 30C45, 30C50

Abstract

In this short note we continue our investigation of nets in locally solid Riesz spaces from [P. Das, E. Savas, {On $\mathcal I$-convergence of nets in locally solid Riesz spaces}, Filomat, 27 (1) (2013), 84--89] and introduce the idea of $\Cal{I}_{\tau}^{\Cal{K}}$-convergence of nets which is more general than $\Cal{I}_{\tau}^*$-convergence and obtain some of its basic properties.

Keywords: Ideal; filter; nets; $\mathcal{I_{\tau}}$-convergence; $\mathcal{I_{\tau}^{\mathcal{K}}}$-convergence; $\mathcal{I_{\tau}^{\mathcal{K}}}$-boundedness; $\mathcal{I}_{\tau}^{\mathcal{K}}$-Cauchy; locally solid Riesz space

MSC: 40G15, 40A35

Abstract

Let $c=|{\mathbb R}|$ denote the cardinality of the continuum and let $\eta$ denote the Euclidean topology on ${\mathbb R}$. Let ${\mathcal L}$ denote the family of all Hausdorff topologies $\tau$ on ${\mathbb R}$ with $\tau\subset\eta$. Let ${\mathcal L}_1$ resp.~${\mathcal L}_2$ resp.~${\mathcal L}_3$ denote the family of all $\tau\in{\mathcal L}$ where $({\mathbb R},\tau)$ is {\it completely normal} resp.~{\it second countable} resp.~{\it not regular}. Trivially, ${\mathcal L}_1\cap{\mathcal L}_3=\emptyset$ and $|{\mathcal L}_i|\leq|{\mathcal L}|\leq 2^c$ and $|{\mathcal L}_2|\leq c$. For $\tau\in{\mathcal L}$ the space $({\mathbb R},\tau)$ is metrizable if and only if $\tau\in{\mathcal L}_1\cap{\mathcal L}_2$. We show that, up to homeomorphism, both ${\mathcal L}_1$ and ${\mathcal L}_3$ contain precisely $2^c$ topologies and ${\mathcal L}_2$ contains precisely $c$ completely metrizable topologies. For $2^c$ non-homeomorphic topologies $\tau\in{\mathcal L}_1$ the space $({\mathbb R},\tau)$ is {\it Baire}, but there are also $2^c$ non-homeomorphic topologies $\tau\in{\mathcal L}_1$ and $c$ non-homeomorphic topologies $\tau\in{\mathcal L}_1\cap{\mathcal L}_2$ where $({\mathbb R},\tau)$ is of {\it first category}. Furthermore, we investigate the {\it complete lattice} ${\mathcal L}_0$ of all topologies $\tau\in{\mathcal L}$ such that $\tau$ and $\eta$ coincide on ${\mathbb R}\setminus\{0\}$. In the lattice ${\mathcal L}_0$ we find $2^c$ (non-homeomorphic) immediate predecessors of the maximum $\eta$, whereas the minimum of ${\mathcal L}_0$ is a compact topology without immediate successors in ${\mathcal L}_0$. We construct chains of homeomorphic topologies in ${\mathcal L}_0\cap{\mathcal L}_1\cap{\mathcal L}_2$ and in ${\mathcal L}_0\cap{\mathcal L}_2\cap{\mathcal L}_3$ and in ${\mathcal L}_0\cap({\mathcal L}_1\setminus{\mathcal L}_2)$ and in ${\mathcal L}_0\cap({\mathcal L}_3\setminus{\mathcal L}_2)$ such that the length of each chain is $c$ (and hence maximal). We also track down a chain in ${\mathcal L}_0$ of length $2^\lambda$ where $\lambda$ is the smallest cardinal number $\kappa$ with $2^\kappa>c$.

Keywords: nonmetrizable Baire spaces; metrizable spaces of first category

MSC: 54A10

Abstract

In this paper, we introduce the notion of radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds giving characterization theorem with some non-trivial examples of such submanifolds. Integrability conditions of distributions $D_1$, $D_2$ and $RadTM$ on radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds have been obtained. Further, we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic.

Keywords: Semi-Riemannian manifold; degenerate metric; radical distribution; screen distribution; screen transversal vector bundle; lightlike transversal vector bundle; Gauss and Weingarten formulae

MSC: 53C15, 53C40, 53C50

Abstract

We consider estimating the extremal index of a maximum autoregressive process of order one under the assumption that the distribution of the innovations has a regularly varying tail at infinity. We establish the asymptotic normality of the new estimator using the extreme quantile approach, and its performance is illustrated in a simulation study. Moreover, we compare, in terms of bias and mean squared error, our estimator with the estimator of Ferro and Segers [Inference for clusters of extreme values, J. Royal Stat. Soc., Ser. B, {65} (2003), 545--556] and Olmo [A new family of consistent and asymptotically-normal estimators for the extremal index, {Econometrics}, 3 (2015), 633--653].

Keywords: extreme value theory; max autoregressive processes; tail index estimation

MSC: 60G70, 62G32

Abstract

We discuss the existence and uniqueness of points of coincidence and common fixed points for a pair of self-mappings defined on a $b$-metric space endowed with a graph. Our results improve and supplement several recent results of metric fixed point theory.

Keywords: $b$-metric; reflexive digraph; point of coincidence; common fixed point

MSC: 47H10, 54H25