Region of variability for functions with positive real part

For \gamma\in\IC such that $|\gamma|<\pi/2$ and $0\leq\beta<1$, let ${\mathcal P}_{\gamma,\beta}$ denote the class of all analytic functions $P$ in the unit disk $\mathbb{D}$ with $P(0)=1$ and {\rm Re\,} \left (e^{i\gamma}P(z)\right)>\beta\cos\gamma \quad \mbox{ in ${\mathbb D}$}. For any fixed $z_0\in\mathbb{D}$ and $\lambda\in\overline{\mathbb{D}}$, we shall determine the region of variability $V_{\mathcal{P}}(z_0,\lambda)$ for $\int_0^{z_0}P(\zeta)\,d\zeta$ when $P$ ranges over the class $$\mathcal{P}(\lambda) = \left\{ P\in{\mathcal P}_{\gamma,\beta} :\, P'(0)=2(1-\beta)\lambda e^{-i\gamma}\cos\gamma \right\}.$$ As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.Comment: 21 pages with 10 figures, to appear in 201

Convolutions of slanted half-plane harmonic mappings

Let ${\mathcal S^0}(H_{\gamma})$ denote the class of all univalent, harmonic, sense-preserving and normalized mappings $f$ of the unit disk \ID onto the slanted half-plane $H_\gamma :=\{w:\,{\rm Re\,}(e^{i\gamma}w) >-1/2\}$ with an additional condition $f_{\bar{z}}(0)=0$. Functions in this class can be constructed by the shear construction due to Clunie and Sheil-Small which allows by examining their conformal counterpart. Unlike the conformal case, convolution of two univalent harmonic convex mappings in \ID is not necessarily even univalent in \ID. In this paper, we fix $f_0\in{\mathcal S^0}(H_{0})$ and show that the convolutions of $f_0$ and some slanted half-plane harmonic mapping are still convex in a particular direction. The results of the paper enhance the interest among harmonic mappings and, in particular, solves an open problem of Dorff, et. al. \cite{DN} in a more general setting. Finally, we present some basic examples of functions and their corresponding convolution functions with specified dilatations, and illustrate them graphically with the help of MATHEMATICA software. These examples explain the behaviour of the image domains.Comment: 15 pages, preprint of December 2011 (submitted to a journal for publication

Cognitive-behavioral treatment of panic disorder with agoraphobia

This paper reports a clinical case study on the effectiveness of Cognitive- Behavioral Treatment (CBT) in treating panic attack with agoraphobia in a local health psychology clinic. M.N., a 24 year old male, complained of nightmares, heart palpitations, sweating, tremors and fearful feelings for thepast one and a half years. He felt anxious about going to crowded places such as bus stations, night markets, supermarkets, and mosques and being left alone in any place which he was not familiar with. This case study adopted an ABC design whereby the subject was assessed at three different phases: pre-treatment, mid-treatment and post-treatment. Four standard assessment measures were administered: Beck Anxiety Inventory (BAI), Beck Depression Inventory (BDI), Anxiety Scale of Minnesota Multiphasic Personality Inventory-2 (MMPI-2) and State-Trait Anxiety Inventory (STAI). The subject responded well to 12 sessions of intervention employed in the study based on CBT model and this could be noticed by minimal score on the entire psychological test administered. The application of behavioral and cognitive strategies became more effective due to patient’s ability to understand and also due to his cooperative behavior. He responded well to imagery exposure and in-vivo gradual exposure and successfully went to shopping malls, used lifts at Kuala Lumpur Tower, went to night markets and used public transport