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Bikash Chakraborty

Publications and source records attributed to Bikash Chakraborty.

32 records · Page 2Linked to original sources
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↗
arXiv2019

Value distribution of some differential monomials

Let $f$ be a transcendental meromorphic function defined in the complex plane $\mathbb{C}$. We consider the value distribution of the differential polynomial $f^{q_{0}}(f^{(k)})^{q_{k}}$, where $q_{0}(\geq 2), q_{k}(\geq 1)$ are $k(\geq1)$ non-negative integers. We obtain a quantitative estimation of the characteristic function $T(r, f)$ in terms of $\overline{N}\left(r,\frac{1}{f^{q_{_{0}}}(f^{(k)})^{q_{k}}-1}\right)$.\par Our result generalizes the results obtained by Xu et al. (Math. Inequal. Appl., 14, 93-100, 2011) and Karmakar and Sahoo (Results Math., 73, 2018) for a particular class of transcendental meromorphic functions.

math.CV↗
arXiv2019

A note on the value distribution of Differential Polynomials

Let $f$ be a transcendental meromorphic function, defined in the complex plane $\mathbb{C}$. In this paper, we give a quantitative estimations of the characteristic function $T(r,f)$ in terms of the counting function of a homogeneous differential polynomial generated by $f$. Our result improves and generalizes some recent results.

math.CV↗
arXiv2018

Some uniqueness results related to the Brück Conjecture

Let f be a non-constant meromorphic function and a = a(z) be a small function of f. Under certain essential conditions, we obtained similar type conclusion of Bruck Conjecture, when f and its differential polynomial P[f] shares a with weight l. Our result improves and generalizes a recent result of Li, Yang, and Liu.

math.CV↗
arXiv2017

A Note On Uniqueness Of Meromorphic Functions And Their Derivatives Sharing Two Sets

In this paper, on the basis of a specific question raised in [6], we further continue our investigations on the uniqueness of a meromorphic function with its higher derivatives sharing two sets and answer the question affirmatively. Moreover, we exhibit some examples to show the sharpness of some conditions used in our main result.

math.CV↗
arXiv2017

Some Aspects of Uniqueness Theory of Entire and Meromorphic Functions (Ph.D. thesis)

The subject of our thesis is the uniqueness theory of meromorphic functions and it is devoted to problems concerning Bruck conjecture, set sharing and related topics. The tool, we used in our discussions is classical Nevanlinna theory of meromorphic functions. In 1996, in order to find the relation between an entire function with its derivative, counterpart sharing one value CM, a famous conjecture was proposed by R. Bruck. Since then the conjecture and its analogous results have been investigated by many researchers and continuous efforts have been put on by them. In our thesis, we have obtained similar types of conclusions as that of Bruck for two differential polynomials which in turn improve several existing results under different sharing environment. A number of examples have been exhibited to justify the necessity or sharpness of some conditions, hypothesis used in the thesis. As a variation of value sharing, F. Gross first introduced the idea of set sharing, by proposing a problem, which has later became popular as Gross Problem. Inspired by the Gross' Problem, the set sharing problems were started which was later shifted towards the characterization of the polynomial backbone of different unique range sets. In our study, we introduced some new type of unique range sets and at the same time, we further explored the anatomy of these unique range sets generating polynomials as well as connected Bruck conjecture with Gross' Problem.

math.CV↗
arXiv2017

Fundamental theorem of Algebra - A Nevanlinna theoretic proof

There are several proofs of the Fundamental Theorem of Algebra, mainly using algebra, analysis and topology. In this article, we have shown that the Fundamental Theorem of Algebra can be proved using Nevanlinna's first fundamental theorem as well as second fundamental theorem also.

math.CV↗
arXiv2017

On some sufficient conditions of strong uniqueness polynomials

In this paper we shall find some sufficient conditions for a uniqueness polynomial to be a strong uniqueness polynomial, as this type of problem was never investigated by researchers earlier.We also exhibit some examples to substantiate our theorems.

math.CV↗