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Bikash Chakraborty

Publications and source records attributed to Bikash Chakraborty.

At least 19 recordsLinked to original sources
arXiv2024

On the cardinality of unique range sets with weight one

Two meromorphic functions $f$ and $g$ are said to share the set $S\subset \mathbb{C}\cup\{\infty\}$ with weight $l\in\mathbb{N}\cup\{0\}\cup\{\infty\}$, if $E_{f}(S,l)=E_{g}(S,l)$ where $$E_{f}(S,l)=\bigcup\limits_{a \in S}\{(z,t) \in \mathbb{C}\times\mathbb{N}~ |~ f(z)=a ~\text{with~ multiplicity}~ p\},$$ where $t=p$ if $p\leq l$ and $t=p+1$ if $p>l$. In this paper, we improve and supplement the result of L. W. Liao and C. C. Yang (On the cardinality of the unique range sets for meromorphic and entire functions, Indian J. Pure appl. Math., 31 (2000), no. 4, 431-440) by showing that there exist a finite set $S$ with cardinality $\geq 13$ such that $E_{f}(S,1)=E_{g}(S,1)$ implies $f\equiv g$.

math.CV↗
arXiv2023

Uniqueness of meromorphic functions that share Two Sets

In this note, we introduce a new kind of pair of finite range sets in $\mathbb{C}$ for meromorphic functions corresponding to their uniqueness, i.e., how two meromorphic functions are uniquely determined by their two finite shared sets.

math.CV↗
arXiv2022

On the Generalizations of Brück Conjecture

We obtain similar types of conclusions as that of Brück [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the necessity or sharpness of some conditions used in the paper. At last, we pose an open problem for future research.

math.CV↗
arXiv2022

Euler's Limit -- Revisited

The aim of this short note is that if $\{ a_{n}\}$ and $\{ b_{n}\}$ are two sequences of positive real numbers such that $a_{n}\to +\infty$ and $b_n$ satisfying the asymptotic formula $b_n\sim k\cdot a_{n}$, where $k>0$, then $\lim\limits_{n\to\infty}\left(1+\frac{1}{a_{n}}\right)^{b_{n}}= e^{k}$.

math.HO↗
arXiv2022

A simple proof of the Fundamental Theorem of Algebra

Many proofs of the Fundamental Theorem of Algebra, including various proofs based on the theory of analytic functions of a complex variable, are known. To the best of our knowledge, this proof is different from the existing ones.

math.GM↗
arXiv2022

A visual tour via the Definite Integration $\int_{a}^{b}\frac{1}{x}dx$

Geometrically, $\int_{a}^{b}\frac{1}{x}dx$ means the area under the curve $\frac{1}{x}$ from $a$ to $b$, where $0<a<b$, and this area gives a positive number. Using this area argument, in this expository note, we present some visual representations of some classical results. For examples, we demonstrate an area argument on a generalization of Euler's limit $\left(\lim\limits_{n\to\infty}\left(\frac{(n+1)}{n}\right)^{n}=e\right)$. Also, in this note, we provide an area argument of the inequality $b^a < a^b$, where $e \leq a< b$, as well as we provide a visual representation of an infinite geometric progression. Moreover, we prove that the Euler's constant $γ\in [\frac{1}{2}, 1)$ and the value of $e$ is near to $2.7$. Some parts of this expository article has been accepted for publication in Resonance - Journal of Science Education, The Mathematical Gazette, and International Journal of Mathematical Education in Science and Technology.

math.HO↗
arXiv2022

Geometric visualizations of $b^{e}<e^{b}$ when $e<b$

In connection to the two fascinating constants $e$ and $π$, there are many beautiful visual proofs to the inequality $π^{e}<e^π$. The aim of this classroom capsule is to give three visual proofs to the more general inequality $b^{e}<e^{b}$ where $e<b$.

math.HO↗
arXiv2021

Unique range sets without Fujimoto's hypothesis

This paper studies the uniqueness of two non-constant meromorphic functions when they share a finite set. Moreover, we will give the existence of unique range sets for meromorphic functions that are zero sets of polynomials that do not necessarily satisfy the Fujimoto's hypothesis.

math.CV↗
arXiv2021

Some results on the unique range sets

In this paper, we exhibit the equivalence between different notions of unique range sets, namely, unique range sets, weighted unique range sets and weak-weighted unique range sets under certain conditions.\par Also, we present some uniqueness theorems which show how two meromorphic functions are uniquely determined by their two finite shared sets. Moreover, in the last section, we make some observations that help us to construct other new classes of unique range sets.

math.CV↗
arXiv2021

Unique range sets of meromorphic functions of non-integer finite order

This paper studies the uniqueness of two non-integral finite ordered meromorphic functions with finitely many poles when they share two finite sets. Also, studies an answer to a question posed by Gross for a particular class of meromorphic functions. Moreover, some observations are made on some results due to Sahoo and Karmakar ( Acta Univ. Sapientiae, Mathematica, DOI: 10.2478/ausm-2018-0025) and Sahoo and Sarkar (Bol. Soc. Mat. Mex., DOI: 10.1007/s40590-019-00260-4).

math.CV↗
arXiv2020

On the value distribution of a Differential Monomial and some normality criteria

Let $f$ be a transcendental meromorphic function defined in the complex plane $\mathbb{C}$, and $φ(\not\equiv 0,\infty)$ be a small function of $f$. In this paper, We give a quantitative estimation of the characteristic function $T(r, f)$ in terms of $N\left(r,\frac{1}{M[f]-φ(z)}\right)$ as well as $\ol{N}\left(r,\frac{1}{M[f]-φ(z)}\right)$, where $M[f]$ is the differential monomial, generated by $f$.\par Moreover, we prove one normality criterion: Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k(\geq1)$, $q_{0}(\geq 3)$, $q_{i}(\geq0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq1)$ be positive integers. If for each $f\in \mathscr{F}$, $f$ has only zeros of multiplicity at least $k$, and $f^{q_{0}}(f')^{q_{1}}...(f^{(k)})^{q_{k}}\not=1$, then $\mathscr{F}$ is normal on domain $D$.

math.CV↗