Sanjay Kumar

• Ph.D.
• Professor at Deen Dayal Upadhyaya College (University of Delhi)

We study about solutions of certain kind of non-linear differential difference equations $$f^{n}(z)+wf^{n-1}(z)f^{'}(z)+f^{(k)}(z+c)=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}$$ and $$f^{n}(z)+wf^{n-1}(z)f^{'}(z)+q(z)e^{Q(z)}f(z+c)=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2} z},$$ where $n\geq 2$, $k\geq0$ are integers, $w, p_{1}, p_{2}, \alpha_{1}$ $\&...

In this paper we study about the existence of solutions of certain kind of non-linear differential and differential-difference equations. We give partial answer to a problem which was asked by chen et al. in [2].

This note investigates the relation between squeezing function and its generalizations. Using the relation obtained, we present an alternate method to find expression of generalized squeezing function of unit ball corresponding to the generalized complex ellipsoids.

In the present article, we define squeezing function corresponding to polydisk and study its properties. We investigate relationship between squeezing function and squeezing function corresponding to polydisk.

We shall address the alternative definition of chain recurrent set for the action of a semigroup of continuous self maps, given by M. Hurley \cite {mh} in noncompact space. Following this, we shall address the characterization of chain recurrence in terms of attractors given by C. Conley in \cite {conley}.

In this paper, we study about existence and non-existence of finite order transcendental entire solutions of the certain non-linear differential-difference equations. We also study about conjectures posed by Rong et al. and Chen et al.

We introduce the notion of squeezing function corresponding to $d$-balanced domains motivated by the concept of generalized squeezing function given by Rong and Yang. In this work we study some of its properties and its relation with Fridman invariant.

We show that all non-trivial solutions of complex differential equation \(f''+ A(z)f'+B(z)f = 0\) are of infinite order if coefficients A(z) and B(z) are of special type and establish a relation between the hyper-order of these solutions and the orders of coefficients A(z) and B(z). We have also extended these results to higher order complex differ...

In the present article, we define squeezing function corresponding to polydisk and study its properties. We investigate relationship between squeezing fuction and squeezing function corresponding to polydisk. We also give an alternate proof for lower bound of the squeezing function of a product domain.

We show that the higher order linear differential equation possesses all solutions of infinite order under certain conditions by extending the work of authors about second order differential equation \cite{dsm2}.

In this paper, we will prove that all non-trivial solutions of $f''+A(z)f'+B(z)f=0$ are of infinite order, where we have some restrictions on entire functions $A(z)$ and $B(z)$.

In this paper, we have considered second order non-homogeneous linear differential equations having entire coefficients. We have established conditions ensuring non-existence of finite order solution of such type of differential equations.

In this paper, we establish transcendental entire function A(z) and polynomial B(z) such that the differential equation f +A(z)f +B(z)f = 0, has all non-trivial solution of infinite order. We use the notion of critical rays of the function e P (z) , where A(z) = d(z)e P (z) with some restrictions.

In this article, we prove a normality criterion for a family of meromorphic functions having zeros with some multiplicity which involves sharing of a holomorphic function by the members of the family. Our result generalizes Montel's normality test in a certain sense.

Schwick, in [6], states that let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$ and if for each $f\in\mathcal{F}$, $(f^n)^{(k)}\neq 1$, for $z\in D$, where $n, k$ are positive integers such that $n\geq k+3$, then $\mathcal{F}$ is a normal family in $D$. In this paper, we investigate the opposite view that if for each $f\in\mathc...

In this paper we prove some normality criteria for a family of meromorphic functions concerning shared analytic functions, which extend or generalized some result obtained by Y. F. Wang, M. L. Fang~\cite{WF} and J. Qui, T. Zhu ~\cite{QZ}.

In this paper we prove some normality criteria for a family of meromorphic functions concerning shared analytic functions, which extend or generalized some result obtained by Y. F. Wang, M. L. Fang [11] and J. Qui, T. Zhu [8].

In this article, we give a Zalcman type renormalization result for the quasinormality of a family of holomorphic functions on a domain in \(\mathbb {C}^n\) that takes values in a complete complex Hermitian manifold.

We have disscussed the problem of finding the condition on coefficients of f + A(z)f + B(z)f = 0, B(z) ≡ 0 so that all non-trivial solutions are of infinite order. The hyper-order of non-trivial solution of infinite order is found when λ(A) < ρ(B) and ρ(B) = ρ(A) or B(z) has Fabry gap.

In this paper, we establish transcendental entire function A(z) and polynomial B(z) such that the differential equation f ′′ + A(z)f ′ + B(z)f = 0, has all non-trivial solution of infinite order. We use the notion of critical rays of the function e P (z) , where A(z) = d(z)e P (z) with some restrictions.

In this paper, we establish transcendental entire function $A(z)$ and polynomial $B(z)$ such that the differential equation $f''+A(z)f'+B(z)f=0$, has all non-trivial solution of infinite order. We use the notion of \emph{critical rays} of the function $e^{P(z)}$, where $A(z)=d(z)e^{P(z)}$ with some restrictions.

For a second order linear differential equation f ′′ + A(z)f ′ + B(z)f = 0, with A(z) and B(z) being transcendental entire functions under some restriction, we have established that all non-trivial solutions are of infinite order. In addition, we have proved that these solutions have infinite number of zeros. Also, we have extended these results to...

For a second order linear differential equation $f''+A(z)f'+B(z)f=0$, with $ A(z)$ and $B(z)$ being transcendental entire functions under some restriction, we have established that all non-trivial solutions are of infinite order. In addition, we have proved that these solutions have infinite number of zeros.

In this paper we prove some normality criteria for a family of meromorphic functions, which involves the zeros of certain differential polynomials generated by the members of the family.

In this article we prove some normality criteria for a family of meromorphic functions which involves sharing of a non-zero value by certain differential monomials generated by the members of the family. These results generalizes some of the results of Schwick.

In this article, we prove a normality criterion for a family of meromorphic functions having zeros with some multiplicity which involves sharing of a holomorphic function by the members of the family. Our result generalizes Montel’s normality test in a certain sense.

Let D be a domain, n, k be positive integers and n >= K+3. Let F be a family of functions meromorphic in D. If each f in F satisfies (f^n)^(k) not equal to 1 for z in D, then F is normal family. This result was proved by Schwick here we give another proof of this result for the case n>=k+1.

Schwick (J Anal Math 52:241–289, 1989) states that let \(\mathcal {F}\) be a family of meromorphic functions on a domain D and if for each \(f\in \mathcal {F}\), \((f^n)^{(k)}\ne 1\), for \(z\in D\), where n, k are positive integers such that \(n\ge k+3\), then \(\mathcal {F}\) is a normal family in D. In this paper we investigate the opposite view...

It is known that the dynamics of f and g vary to a large extent from that of its composite entire functions. We have shown using Approximation theory of entire functions, the existence of entire functions f and g having infinite number of domains satisfying various properties and relating it to their composition. We have explored and enlarged all t...

In this paper, we have shown that, by using results of Aladro and Krantz and of Fujimoto, Zalcman's type Lemma can be given for quasinormality of a family of holomorphic functions on a domain of $\mathbb{C}^n$ into a complete complex Hermitian manifold.

We discuss the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory and provide some condition for the complete invariance of escaping set and Julia set of transcendental semigroups. Some results on limit functions and postsingular set have been discussed. A class of hyperbolic transcendental semigroups and...

We study the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some conditions for connectivity of the Julia set of the transcendental semigroups. We...

We study the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some conditions for connectivity of the Julia set of the transcendental semigroups. We...

In this article, we prove a normality criterion for a family of meromorphic functions which involves sharing of holomorphic functions. Our result generalizes some of the results of H. H. Chen, M. L. Fang and M. Han, Y. Gu.

We consider the dynamical properties of transcendental entire functions and their compositions. We give several conditions under which Fatou set of a transcendental entire function $f$ coincide with that of $f\circ g,$ where $g$ is another transcendental entire function. We also prove some result giving relationship between singular values of trans...

In this article, we prove some normality criteria for a family of meromorphic functions having multiple zeros and poles which involves sharing of a non-zero value by zeros and poles which involves a non linear differential polynomial.

In this paper, we obtained some normality criteria for families of holomorphic functions. Which generalizes some results of Fang, Xu, Chen and Hua.

We establish a criterion for local boundedness and hence normality of a family $\F$ of analytic functions on a domain $D$ in the complex plane whose corresponding family of derivatives is locally bounded. Furthermore we investigate the relation between domains of normality of a family $\F$ of meromorphic functions and its corresponding Schwarzian d...

In this paper, we obtain common fixed point theorems of weakly compatible maps on symmetric spaces. We prove that, if S and T are weakly compatible maps satisfying property (E-A) along with strict contractive conditions, then they have common fixed points. Since these results are obtained without using the full force of a metric, they are improved...