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Area distortion under meromorphic mappings with nonzero pole having quasiconformal extension
Bappaditya Bhowmik,Goutam Satpati 대한수학회 2019 대한수학회지 Vol.56 No.2
Let $\Sigma_k(p)$ be the class of univalent meromorphic functions defined on the unit disc $\mathbb D$ with $k$-quasiconformal extension to the extended complex plane $\widehat{\mathbb C}$, where $0\leq k < 1$. Let $\Sigma_k^0(p)$ be the class of functions $f \in \Sigma_k(p)$ having expansion of the form $f(z)= 1/(z-p) + \sum_{n=1}^{\infty}b_n z^{n}$ on $\mathbb D.$ In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in $\Sigma_k^0(p)$. As a consequence of the obtained results, we present a sharp upper bound for the Hilbert transform of characteristic function of a Lebesgue measurable subset of $\mathbb D$.
SUFFICIENT CONDITIONS FOR UNIVALENCE AND STUDY OF A CLASS OF MEROMORPHIC UNIVALENT FUNCTIONS
Bhowmik, Bappaditya,Parveen, Firdoshi Korean Mathematical Society 2018 대한수학회보 Vol.55 No.3
In this article we consider the class ${\mathcal{A}}(p)$ which consists of functions that are meromorphic in the unit disc $\mathbb{D}$ having a simple pole at $z=p{\in}(0,1)$ with the normalization $f(0)=0=f^{\prime}(0)-1$. First we prove some sufficient conditions for univalence of such functions in $\mathbb{D}$. One of these conditions enable us to consider the class ${\mathcal{A}}_p({\lambda})$ that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish that ${\mathcal{U}}_p({\lambda}){\subsetneq}{\mathcal{A}}_p({\lambda})$, where ${\mathcal{U}}_p({\lambda})$ was introduced and studied in [2]. Finally, we discuss some coefficient problems for ${\mathcal{A}}_p({\lambda})$ and end the article with a coefficient conjecture.
Coefficient discs and generalized central functions for the class of concave schlicht functions
Bappaditya Bhowmik,Karl-Joachim Wirths 대한수학회 2014 대한수학회보 Vol.51 No.5
We consider functions that map the open unit disc confor- mally onto the complement of an unbounded convex set with opening angle πα, α ∈ (1, 2], at infinity. We derive the exact interval for the vari- ability of the real Taylor coefficients of these functions and we prove that the corresponding complex Taylor coefficients of such functions are con- tained in certain discs lying in the right half plane. In addition, we also determine generalized central functions for the aforesaid class of functions.
AREA DISTORTION UNDER MEROMORPHIC MAPPINGS WITH NONZERO POLE HAVING QUASICONFORMAL EXTENSION
Bhowmik, Bappaditya,Satpati, Goutam Korean Mathematical Society 2019 대한수학회지 Vol.56 No.2
Let ${\Sigma}_k(p)$ be the class of univalent meromorphic functions defined on the unit disc ${\mathbb{D}}$ with k-quasiconformal extension to the extended complex plane ${\hat{\mathbb{C}}}$, where $0{\leq}k<1$. Let ${\Sigma}^0_k(p)$ be the class of functions $f{\in}{\Sigma}_k(p)$ having expansion of the form $f(z)=1/(z-p)+{\sum_{n=1}^{\infty}}\;b_nz^n$ on ${\mathbb{D}}$. In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in ${\sum_{k}^{0}}(p)$. As a consequence of the obtained results, we present a sharp upper bound for the Hilbert transform of characteristic function of a Lebesgue measurable subset of ${\mathbb{D}}$.
COEFFICIENT DISCS AND GENERALIZED CENTRAL FUNCTIONS FOR THE CLASS OF CONCAVE SCHLICHT FUNCTIONS
Bhowmik, Bappaditya,Wirths, Karl-Joachim Korean Mathematical Society 2014 대한수학회보 Vol.51 No.5
We consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle ${\pi}{\alpha}$, ${\alpha}{\in}(1,2]$, at infinity. We derive the exact interval for the variability of the real Taylor coefficients of these functions and we prove that the corresponding complex Taylor coefficients of such functions are contained in certain discs lying in the right half plane. In addition, we also determine generalized central functions for the aforesaid class of functions.
Sufficient conditions for univalence and study of a class of meromorphic univalent functions
Bappaditya Bhowmik,Firdoshi Parveen 대한수학회 2018 대한수학회보 Vol.55 No.3
In this article we consider the class $\mathcal{A}(p)$ which consists of functions that are meromorphic in the unit disc $\ID$ having a simple pole at $z=p\in (0,1)$ with the normalization $f(0)=0=f'(0)-1 $. First we prove some sufficient conditions for univalence of such functions in $\ID$. One of these conditions enable us to consider the class $\mathcal{V}_{p}(\lambda)$ that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish that $\mathcal{U}_{p}(\lambda)\subsetneq \mathcal{V}_{p}(\lambda)$, where $\mathcal{U}_{p}(\lambda)$ was introduced and studied in \cite{BF-1}. Finally, we discuss some coefficient problems for $\mathcal{V}_{p}(\lambda)$ and end the article with a coefficient conjecture.