arXiv subjects
Publications and source records attributed to S. Ponnusamy.
We consider the class of meromorphic univalent functions having a simple pole at $p\in(0,1)$ and that map the unit disc onto the exterior of a domain which is starlike with respect to a point $w_0 \neq 0,\, \infty$. We denote this class of functions by $Σ^*(p,w_0)$. In this paper, we find the exact region of variability for the second Taylor coefficient for functions in $Σ^*(p,w_0)$. In view of this result we rectify some results of James Miller.
Let $\ID$ denote the open unit disc and let $p\in (0,1)$. We consider the family $Co(p)$ of functions $f:\ID\to \overline{\IC}$ that satisfy the following conditions: \bee \item[(i)] $f$ is meromorphic in $\ID$ and has a simple pole at the point $p$. \item[(ii)] $f(0)=f'(0)-1=0$. \item[(iii)] $f$ maps $\ID$ conformally onto a set whose complement with respect to $\overline{\IC}$ is convex. \eee We determine the exact domains of variability of some coefficients $a_n(f)$ of the Laurent expansion $$f(z)=\sum_{n=-1}^{\infty} a_n(f)(z-p)^n,\quad |z-p|<1-p, $$ for $f\in Co(p)$ and certain values of $p$. Knowledge on these Laurent coefficients is used to disprove a conjecture of the third author on the closed convex hull of $Co(p)$ for certain values of $p$.
Let $Co(α)$ denote the class of concave univalent functions in the unit disk $\ID$. Each function $f\in Co(α)$ maps the unit disk $\ID$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional $(1-|z|^2)\left ( f''(z)/f'(z)\right)$, $f\in Co(α)$. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional $(1-|z|^2)\left(f''(z)/f'(z)\right)$, $f\in Co(α)$ whenever $f''(0)$ is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in $Co(α)$ belong to the $H^p$ space for $p<1/α$.
For $γ\in\IC$ such that $|γ|<π/2$ and $0\leqβ<1$, let ${\mathcal P}_{γ,β} $ denote the class of all analytic functions $P$ in the unit disk $\mathbb{D}$ with $P(0)=1$ and $$ {\rm Re\,} \left (e^{iγ}P(z)\right)>β\cosγ\quad \mbox{ in ${\mathbb D}$}. $$ For any fixed $z_0\in\mathbb{D}$ and $λ\in\overline{\mathbb{D}}$, we shall determine the region of variability $V_{\mathcal{P}}(z_0,λ)$ for $\int_0^{z_0}P(ζ)\,dζ$ when $P$ ranges over the class $$ \mathcal{P}(λ) = \left\{ P\in{\mathcal P}_{γ,β} :\, P'(0)=2(1-β)λe^{-iγ}\cosγ\right\}. $$ 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.
For $α\in\IC\setminus \{0\}$ let $\mathcal{E}(α)$ denote the class of all univalent functions $f$ in the unit disk $\mathbb{D}$ and is given by $f(z)=z+a_2z^2+a_3z^3+\cdots$, satisfying $$ {\rm Re\,} \left (1+ \frac{zf''(z)}{f'(z)}+αzf'(z)\right)>0 \quad {in ${\mathbb D}$}. $$ For any fixed $z_0$ in the unit disk $\mathbb{D}$ and $λ\in\overline{\mathbb{D}}$, we determine the region of variability $V(z_0,λ)$ for $\log f'(z_0)+αf(z_0)$ when $f$ ranges over the class $$\mathcal{F}_α(λ)=\left\{f\in\mathcal{E}(α) \colon f''(0)=2λ-α%\quad{and} f'''(0)=2[(1-|λ|^2)a+ %(λ-α)^2 -λα] \right\}. $$ We geometrically illustrate the region of variability $V(z_0,λ)$ for several sets of parameters using Mathematica. In the final section of this article we propose some open problems.
In this paper we determine the region of variability for spirallike funcions with respect to a boundary point. In the final section we graphically illustrate the region of variability for several sets of parameters.
In this paper we determine the region of variability for certain subclasses of univalent functions satisfying differential inequalities. In the final section we graphically illustrate the region of variability for several sets of parameters.