Abstract

For analytic function $f(z)=z+a_2z^2+\cdots$ in the open unit disc $\mathbb{D}$, a new fractional operator $\mbox{D}^\lambda f(z)$ is defined. Applying this fractional operator $\mbox{D}^\lambda f(z)$ and the principle of subordination, we give new proofs for some classical results concerning the class $\cal{S}_\lambda^*(A,B,\alpha)$ of functions $f(z)$.

Keywords: Starlike, fractional integral, fractional derivative, distortion theorem.

MSC: 30C45

Abstract

A space $X$ is $C$-Lindel{ö}f (weakly $C$-Lindel{ö}f) if for every closed subset $F$ of $X$ and every open cover $\cal U$ of $F$ by open subsets of $X$, there exists a countable subfamily $\cal V$ of $\cal U$ such that $F\subseteq \cup\{\overline V:V \in\cal V\}$ (respectively, $F\subseteq \overline{\cup \cal V}$). In this paper, we investigate the relationships among $C$-Lindel{ö}f spaces, weakly $C$-Lindel{ö}f spaces and Lindel{ö}f spaces, and also study various properties of weakly $C$-Lindel{ö}f spaces and $C$-Lindel{ö}f spaces.

Keywords: Lindelöf space, $C$-Lindelöf space, weakly $C$-Lindelöf space.

MSC: 54D15, 54D20

Abstract

In this paper, we give some characterizations of almost open mappings defined on first countable spaces, which corrects some errors for almost open mappings.

Keywords: Almost open mapping, almost weak-open mapping, almost $sn$-open mapping, 1-sequence-covering mapping.

MSC: 54C05, 54C10

Abstract

In this paper, we obtain some results concerning the scores in oriented graphs. Further, we give a new and direct proof of the Theorem on oriented graph scores due to Avery.

Keywords: Tournament, oriented graph, score sequence, triple, transitive.

MSC: 05C20

Abstract

We will give a necessary and sufficient condition for the infinitesimal generator of a strongly continuous cosine operator function $C(t)$, such that $\|C(t)\|\le1$ for all $t\in R$ on a reflexive, strictly convex (complex) Banach space with a G\^ateaux differentiable norm to be a spectral scalar type operator with the spectral family of hermitian bounded linear projectors.

Keywords: Cosine operator function, hermitian operator, strictly convex Banach space, G\^ateaux differentiable norm, spectral operator.

MSC: 47D09, 46C50, 47B40

Abstract

A class of $p$-valent harmonic functions associated with a certain generalized Ruscheweyh type operator is introduced. Among the various properties investigated for this class of functions are the results giving the coefficient bounds, distortion properties and extreme points.

Keywords: Multivalent functions, harmonic functions, distortion bounds, extreme points, Ruscheweyh type operator.

MSC: 30C45

Abstract

In this paper, compact composition operators acting on Hardy-Orlicz spaces $$H^{\Phi} = \big\{\, f \in H({\Bbb D}) : \sup_{0 < r < 1} \int_{\partial {\Bbb D}} \Phi(\log^{+} |f(r e^{i \theta})|)\, d \sigma < \infty \,\big\} $$ are studied. In fact, we prove that if $\Phi$ is a twice differentiable, non-constant, non-decreasing non-negative, convex function on $\Bbb R$, then the composition operator $C_{\varphi}$ induced by a holomorphic self-map $\varphi$ of the unit disk is compact on Hardy-Orlicz spaces $H^{\Phi}$ if and only if it is compact on the Hardy space $H^{2}$.

Keywords: Hardy-Orlicz space, Composition operator, Nevanlinna counting function, vanishing Carleson measure.

MSC: 47B33, 46E38, 30D55

Abstract

We prove a uniqueness theorem for meromorphic functions sharing three weighted values, as consequences of which a number of results follow.

Keywords: Meromorphic function, weighted sharing, uniqueness.

MSC: 30D35