On the Fekete-Szegö problem for concave univalent functions
We consider the Fekete-Szegö problem with real parameter $λ$ for the class $Co(α)$ of concave univalent functions.
math.CV↗
arXiv subjects
B. Bhowmik
Publications and source records attributed to B. Bhowmik.
On some problems of James Miller
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.
math.CV↗
Domains of variability of Laurent coefficients and the convex hull for the family of concave univalent functions
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 $p$.
math.CV↗
Characterization and the pre-Schwarzian norm estimate for concave univalent functions
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 $p<1/α$.
math.CV↗