Dinesh Kumar’s research while affiliated with University of Delhi and other places

... Then, there exists a positive integer k satisfying 2π n = k 2 m+2 , that is, m + 2 = kn, which contradicts n ∤ m + 2. Thus, we complete the proof. □ Theorem 2 is motivated by Theorem C given by Kumar and Saini [12]. They considered, A(z) has Fabry gaps and ρ(B) < ρ(A). ...

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Study of Growth of Certain Second Order Linear Differential Equations

... The motivation for the proof comes from [13] Let's consider two commuting functions of class B i.e. f • g = g • f . Functions of class B have the property that J(f ) = J(g) and therefore J(f • g) = J(f ) [10], which implies that F (f • g) = F (f ) = F (g). ...

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Results on Dynamics of Bungee set of Composite Entire Functions in the Eremenko-Lyubich Class

... The conditions on the coe‰cients of equation (1) have been studied so that all solutions of (1) are of infinite order [2,16,17]. In [7,8,9,10,11], certain conditions have been established on the coe‰cients of associated homogeneous di¤erential equation (2) for the existence of non-trivial solutions of infinite order. We note here that if all non-trivial solutions of equation (2) are of infinite order under certain conditions, then it may happen that there is a finite order solution of equation (1) under the same conditions. ...

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The growth of solutions of non-homogeneous linear differential equations

... Also, they [6] proved that for any transcendental semigroup, it is forward invariant and later Kumar et.al. [7] proved that for any abelian transcendental semigroup escaping set is backward invariant. Also, they [7] proved that J(H) ∩ I (H) = ∅ and J(H) = ∂I (H) where J(H) is Julia set of transcendental entire semigroup H. Later, Subedi [5] introduced another version of escaping set of holomorphic semigroup. ...

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Escaping set and Bounded Orbit Set of Holomorphic Semigroup on C

... In the second section, we have extended the results of nearly abelian rational semigroups to the best possible class of nearly abelian transcendental semigroups. Moreover, we have extended some of the results of [13,14,15] to nearly abelian transcendental semigroups. ...

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Dynamics of Nearly Abelian Transcendental Semigroup

... The motivation for the proof comes from [13] Let's consider two commuting functions of class B i.e. f • g = g • f . Functions of class B have the property that J(f ) = J(g) and therefore J(f • g) = J(f ) [10], which implies that F (f • g) = F (f ) = F (g). This implies that a point z 0 ∈ J(f • g) can't belong to F (g) (or F (f )), because that would imply that z 0 ∈ F (f • g) as well, and that's a clear contradiction (since I(f • g) is contained in J(f )). ...

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Results on Dynamics of Bungee set of Composite Entire Functions in the Eremenko-Lyubich Class

... Recall that if g and h are transcendental entire functions and f is a continuous map of the complex plane into itself with f @BULLET g = h @BULLET f, then g and h are said to be semiconjugated by f and f is called a semiconjugacy [5]. In [11], the first author considered the dynamics of semiconjugated entire functions and provided several conditions under which the semiconjugacy carries Fatou set of one entire function into Fatou set of other entire function appearing in the semiconjugation. Furthermore, it was shown that under certain conditions on the growth of entire functions appearing in the semiconjugation, the set of asymptotic values of the derivative of composition of the entire functions is bounded. ...

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107-114 Kumar HRI 631

... Lemma 3.7 (Theorem 2.1(ii) of [15]). Let f and g be permutable transcendental entire functions. ...

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Semigroups of transcendental entire functions and their dynamics

... Singh [2] constructed examples of composition of transcendental entire functions whose dynamics is different from its individual factors. This result was further extended to infinitely many such domains in [11]. Many authors have studied existence of wandering domains in rather different ways. ...

Reference:

On the Dynamics of Composition of Transcendental Entire Functions in Angular Region. Here is the link: https://rdcu.be/bGbyc