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Sumit Nagpal

Publications and source records attributed to Sumit Nagpal.

14 recordsLinked to original sources
arXiv2021

Geometric properties of a domain with cusps

For $n\geq 4$ (even), the function $φ_{n\mathcal{L}}(z)=1+nz/(n+1)+z^n/(n+1)$ maps the unit disk $\mathbb{D}$ onto a domain bounded by an epicycloid with $n-1$ cusps. In this paper, the class $\mathcal{S}^*_{n\mathcal{L}} = \mathcal{S}^*(φ_{n\mathcal{L}})$ is studied and various inclusion relations are established with other subclasses of starlike functions. The bounds on initial coefficients is also computed. Various radii problems are also solved for the class $\mathcal{S}^*_{n\mathcal{L}}$.

math.CV↗
arXiv2021

Marx-Strohhäcker theorem for Multivalent Functions

Some differential implications of classical Marx-Strohhäcker theorem are extended for multivalent functions. These results are also generalized for functions with fixed second coefficient by using the theory of first order differential subordination which in turn, corrects the results of Selvaraj and Stelin [On multivalent functions associated with fixed second coefficient and the principle of subordination, Int. J. Math. Anal. {\bf 9} (2015), no.~18, 883--895].

math.CV↗
arXiv2021

Geometric Properties of Generalized Bessel Function associated with the Exponential Function

Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implications are derived for analytic functions involving Bessel function and the operator introduced by Baricz \emph{et al.} [Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc. {\bf 38} (2015), no.~3, 1255--1280]. These results are obtained by constructing suitable class of admissible functions. Examples involving trigonometric and hyperbolic functions are provided to illustrate the obtained results.

math.CV↗
arXiv2020

Inclusion relations and radius problems for a subclass of starlike functions

By considering the polynomial function $ϕ_{car}(z)=1+z+z^2/2,$ we define the class $\Scar$ consisting of normalized analytic functions $f$ such that $zf'/f$ is subordinate to $ϕ_{car}$ in the unit disk. The inclusion relations and various radii constants associated with the class $\Scar$ and its connection with several well-known subclasses of starlike functions is established. As an application, the obtained results are applied to derive the properties of the partial sums and convolution.

math.CV↗
arXiv2019

Exponential starlikeness and convexity of confluent hypergeometric, Lommel and Struve functions

Sufficient conditions are obtained on the parameters of Lommel function of the first kind, generalized Struve function of the first kind and the confluent hypergeometric function under which these special functions become exponential convex and exponential starlike in the open unit disk. The method of differential subordination is employed in proving the results. Few examples are also provided to illustrate the results obtained.

math.CV↗
arXiv2019

Starlikeness Associated With The Exponential Function

Given a domain $Ω$ in the complex plane $\mathbb{C}$ and a univalent function $q$ defined in an open unit disk $\mathbb{D}$ with nice boundary behaviour, Miller and Mocanu studied the class of admissible functions $Ψ(Ω,q)$ so that the differential subordination $ψ(p(z),zp(z),z^2p''(z);z)\prec h(z)$ implies $p(z)\prec q(z)$ where $p$ is an analytic function in $\mathbb{D}$ with $p(0)=1$, $ψ:\mathbb{C}^3\times \mathbb{D}\to\mathbb{C}$ and $Ω=h(\mathbb{D})$. This paper investigates the properties of this class for $q(z)=e^z$. As application, several sufficient conditions for normalized analytic functions $f$ to be in the subclass of starlike functions associated with the exponential function are obtained.

math.CV↗
arXiv2013

Starlikeness, convexity and close-to-convexity of harmonic mappings

In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic function whose analytic part is convex, is univalent and close-to-convex. In this paper, certain cases are discussed under which the conclusion of this result can be strengthened and extended to fully starlike and fully convex harmonic mappings. In addition, we investgate the properties of functions in the class $\mathcal{M}(α)$ $(|α|\leq 1)$ consisting of harmonic functions $f=h+\overline{g}$ with $g'(z)=αzh'(z)$, $\RE (1+{zh''(z)}/{h'(z)})>-{1}/{2} $ $ \mbox{for} |z|<1 $. The coefficient estimates, growth results, area theorem and bounds for the radius of starlikeness and convexity of the class $\mathcal{M}(α)$ are determined. In particular, the bound for the radius of convexity is sharp for the class $\mathcal{M}(1)$.

math.CV↗
arXiv2013

A comprehensive class of harmonic functions defined by convolution and its connection with integral transforms and hypergeometric functions

For given two harmonic functions $Φ$ and $Ψ$ with real coefficients in the open unit disk $\mathbb{D}$, we study a class of harmonic functions $f(z)=z-\sum_{n=2}^{\infty}A_nz^{n}+\sum_{n=1}^{\infty}B_n\bar{z}^n$ $(A_n, B_n \geq 0)$ satisfying \[\RE \frac{(f*Φ)(z)}{(f*Ψ)(z)}>α\quad (0\leq α<1, z \in \mathbb{D});\] * being the harmonic convolution. Coefficient inequalities, growth and covering theorems, as well as closure theorems are determined. The results obtained extend several known results as special cases. In addition, we study the class of harmonic functions $f$ that satisfy $\RE f(z)/z>α$ $(0\leq α<1, z \in \mathbb{D})$. As an application, their connection with certain integral transforms and hypergeometric functions is established.

math.CV↗
arXiv2013

Univalence and convexity in one direction of the convolution of harmonic mappings

Let $\mathcal{H}$ denote the class of all complex-valued harmonic functions $f$ in the open unit disk normalized by $f(0)=0=f_{z}(0)-1=f_{\bar{z}}(0)$, and let $\mathcal{A}$ be the subclass of $\mathcal{H}$ consisting of normalized analytic functions. For $ϕ\in \mathcal{A}$, let $\mathcal{W}_{H}^{-}(ϕ):=\{f=h+\bar{g} \in \mathcal{H}:h-g=ϕ\}$ and $\mathcal{W}_{H}^{+}(ϕ):=\{f=h+\bar{g} \in \mathcal{H}:h+g=ϕ\}$ be subfamilies of $\mathcal{H}$. In this paper, we shall determine the conditions under which the harmonic convolution $f_1*f_2$ is univalent and convex in one direction if $f_1 \in \mathcal{W}_{H}^{-}(z)$ and $f_2 \in \mathcal{W}_{H}^{-}(ϕ)$. A similar analysis is carried out if $f_1 \in \mathcal{W}_{H}^{-}(z)$ and $f_2 \in \mathcal{W}_{H}^{+}(ϕ)$. Examples of univalent harmonic mappings constructed by way of convolution are also presented.

math.CV↗
arXiv2012

Construction of subclasses of univalent harmonic mappings

Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk are widely studied. A new methodology is employed to construct subclasses of univalent harmonic mappings from a given subfamily of univalent analytic functions. The notion of harmonic Alexander integral operator is introduced. Also, the radius of convexity for certain families of harmonic functions is determined.

math.CV↗
arXiv2012

Second-order differential subordinations for analytic functions with fixed initial coefficient

Functions with fixed initial coefficient have been widely studied. A new methodology is proposed in this paper by making appropriate modifications and improvements to the theory of second-order differential subordination. Several interesting examples are given. The results obtained are applied to the classes of convex and starlike functions with fixed second coefficient.

math.CV↗
arXiv2012

Fully Starlike and Convex Harmonic Mappings of order α

The hereditary property of convexity and starlikeness for conformal mappings does not generalize to univalent harmonic mappings. This failure leads us to the notion of fully starlike and convex mappings of order α, (0\leq α<1). A bound for the radius of fully starlikeness and fully convexity of order αis determined for certain families of univalent harmonic mappings. Convexity is not preserved under the convolution of univalent harmonic convex mappings, unlike in the analytic case. Given two univalent harmonic convex mappings f and g, the problem of finding the radius r_{0} such that f*g is a univalent harmonic convex mapping in |z|<r_{0}, is being considered.

math.CV↗
arXiv2012

A subclass of close-to-convex harmonic mappings

A subclass of complex-valued close-to-convex harmonic functions that are univalent and sense-preserving in the open unit disc is investigated. The coefficient estimates, growth results, area theorem, boundary behavior, convolution and convex combination properties for the above family of harmonic functions are obtained.

math.CV↗