arXiv subjects

V. Ravichandran

Publications and source records attributed to V. Ravichandran.

At least 19 recordsLinked to original sources

arXiv2022

Analysis of Validating and Verifying OpenACC Compilers 3.0 and Above

OpenACC is a high-level directive-based parallel programming model that can manage the sophistication of heterogeneity in architectures and abstract it from the users. The portability of the model across CPUs and accelerators has gained the model a wide variety of users. This means it is also crucial to analyze the reliability of the compilers' implementations. To address this challenge, the OpenACC Validation and Verification team has proposed a validation testsuite to verify the OpenACC implementations across various compilers with an infrastructure for a more streamlined execution. This paper will cover the following aspects: (a) the new developments since the last publication on the testsuite, (b) outline the use of the infrastructure, (c) discuss tests that highlight our workflow process, (d) analyze the results from executing the testsuite on various systems, and (e) outline future developments.

cs.SE↗

arXiv2022

Starlikeness of a product of starlike functions with non-vanishing polynomials

For a function $f$ starlike of order $α$, $0\leqslant α<1$, a non-constant polynomial $Q$ of degree $n$ which is non-vanishing in the unit disc $\mathbb{D}$ and $β>0$, we consider the function $F:\mathbb{D}\to\mathbb{C}$ defined by $F(z)=f(z) (Q(z))^{β/n}$ and find the largest value of $r\in (0,1]$ such that $r^{-1} F(rz)$ lies in various known subclasses of starlike functions such as the class of starlike functions of order $λ$, the classes of starlike functions associated with the exponential function, cardioid, a rational function, nephroid domain and modified sigmoid function. Our radii results are sharp. We also discuss the correlation with known radii results as special cases.

math.CV↗

arXiv2022

The Booth Lemniscate Starlikeness Radius for Janowski Starlike Functions

The function $G_α(z)=1+ z/(1-αz^2)$, \, $0\leq α<1$, maps the open unit disc $\mathbb{D}$ onto the interior of a domain known as the Booth lemniscate. Associated with this function $G_α$ is the recently introduced class $\mathcal{BS}(α)$ consisting of normalized analytic functions $f$ on $\mathbb{D}$ satisfying the subordination $zf'(z)/f(z) \prec G_α(z)$. Of interest is its connection with known classes $\mathcal{M}$ of functions in the sense $g(z)=(1/r)f(rz)$ belongs to $\mathcal{BS}(α)$ for some $r$ in $(0,1)$ and all $f \in \mathcal{M}$. We find the largest radius $r$ for different classes $\mathcal{M}$, particularly when $\mathcal{M}$ is the class of starlike functions of order $β$, or the Janowski class of starlike functions. As a primary tool for this purpose, we find the radius of the largest disc contained in $G_α(\mathbb{D})$ and centered at a certain point $a \in \mathbb{R}$.

math.CV↗

arXiv2022

Criteria for Starlikeness Using Schwarzian Derivatives

For a normalised analytic function f defined on the open unit disk in the complex plane, we determine several sufficient conditions for starlikeness in terms of the quotients Q_{ST}:=zf'(z)/f(z), Q_{CV}:=1+zf"(z)/f'(z) and the Schwarzian derivative Q_{SD}:=z^2((f"(z)/f'(z))'-(f"(z)/f'(z))^2/2)$. These conditions were obtained by using the admissibility criteria of starlikeness in the theory of second order differential subordination.

math.CV↗

arXiv2021

Toeplitz determinants associated with Ma-Minda classes of starlike and convex functions

A starlike function $f$ is characterized by the quantity $zf'(z)/f(z)$ lying in the right half-plane. This paper deals with sharp bounds for certain symmetric Toeplitz determinants whose entries are the coefficients of the functions $f$ for which the quantity $zf'(z)/f(z)$ takes values in certain specific subset in the right half-plane. The results obtained include several new special cases and some known results.

math.CV↗

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

Briot-Bouquet differential subordination and Bernardi's integral operator

The conditions on $A$, $B$, $β$ and $γ$ are obtained for an analytic function $p$ defined on the open unit disc $\mathbb{D}$ and normalized by $p(0)=1$ to be subordinate to $(1+Az)/(1+Bz)$, $-1\leq B<A \leq 1$ when $p(z)+ zp'(z)/(βp(z)+γ)$ is subordinate to $e^{z}$. The conditions on these parameters are derived for the function $p$ to be subordinate to $\sqrt{1+z}$ or $e^{z}$ when $p(z)+ zp'(z)/(βp(z)+γ)$ is subordinate to $(1+Az)/(1+Bz)$. The conditions on $β$ and $γ$ are determined for the function $p$ to be subordinate to $e^{z}$ when $p(z)+ zp'(z)/(βp(z)+γ)$ is subordinate to $\sqrt{1+z}$. Related result for the function $p(z)+ zp'(z)/(βp(z)+γ)$ to be in the parabolic region bounded by the $\operatorname{Re} w=|w-1|$ is investigated. Sufficient conditions for the Bernardi's integral operator to belong to the various subclasses of starlike functions are obtained as applications

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↗

arXiv2021

Starlikeness of Analytic Functions with Subordinate Ratios

Let $h$ be a non-vanishing analytic function in the open unit disc with $h(0)=1$. Consider the class consisting of normalized analytic functions $f$ whose ratios $f(z)/g(z)$, $g(z)/z p(z)$, and $p(z)$ are each subordinate to $h$ for some analytic functions $g$ and $p$. The radius of starlikeness is obtained for this class when $h$ is chosen to be either $h(z)=\sqrt{1+z}$ or $h(z)=e^z$. Further $\mathcal{G}$-radius is also obtained for each of these two classes when $\mathcal{G}$ is a particular widely studied subclass of starlike functions. These include $\mathcal{G}$ consisting of the Janowski starlike functions, and functions which are parabolic starlike.

math.CV↗

arXiv2021

Radius of starlikeness for some classes containing non-univalent functions

A starlike univalent function $f$ is characterized by the function $zf'(z)/f(z)$; several subclasses of these functions were studied in the past by restricting the function $zf'(z)/f(z)$ to take values in a region $Ω$ on the right-half plane, or, equivalently, by requiring the function $zf'(z)/f(z)$ to be subordinate to the corresponding mapping of the unit disk $\mathbb{D}$ to the region $Ω$. The mappings $w_1(z):=z+\sqrt{1+z^2}, w_2(z):=\sqrt{1+z}$ and $w_3(z):=e^z$ maps the unit disk $\mathbb{D}$ to various regions in the right half plane. For normalized analytic functions $f$ satisfying the conditions that $f(z)/g(z), g(z)/zp(z)$ and $p(z)$ are subordinate to the functions $w_i, i=1,2,3$ in various ways for some analytic functions $g(z)$ and $p(z)$, we determine the sharp radius for them to belong to various subclasses of starlike functions.

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↗

arXiv2020

Directional Convexity of Combinations of Harmonic Half-Plane and Strip Mappings

For $k=1,2$, let $f_k=h_k+\overline{g_k}$ be normalized harmonic right half-plane or vertical strip mappings. We consider the convex combination $\hat{f}=ηf_1+(1-η)f_2 =ηh_1+(1-η)h_2 +\overline{\overlineη g_1+(1-\overlineη)g_2}$ and the combination $\tilde{f}=ηh_1+(1-η)h_2+\overline{ηg_1+(1-η)g_2}$. For real $η$, the two mappings $\hat{f}$ and $\tilde{f}$ are the same. We investigate the univalence and directional convexity of $\hat{f}$ and $\tilde{f}$ for $η\in\mathbb{C}$. Some sufficient conditions are found for convexity of the combination $\tilde{f}$.

math.CV↗

arXiv2020

Radius of Starlikeness for Bloch Functions

For normalised analytic functions $f$ defined on the open unit disc $\mathbb{D}$ satisfying the condition $\sup_{z\in \mathbb{D}}(1-|z^2|) |f'(z)|\leq 1$, known as Bloch functions, we determine various starlikeness radii.

math.CV↗

arXiv2020

Starlikeness of Certain Non-Univalent Functions

We consider three classes of functions defined using the class $\mathcal{P}$ of all analytic functions $p(z)=1+cz+\dotsb$ on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions $f$ with $f/g\in\mathcal{P}$ and $g/(zp)\in\mathcal{P}$ for some normalized analytic function $g$ and $p\in \mathcal{P}$. The second class is defined by replacing the condition $f/g\in\mathcal{P}$ by $|(f/g)-1|<1$ while the other class consists of normalized analytic functions $f$ with $f/(zp)\in\mathcal{P}$ for some $p\in \mathcal{P}$. We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order $α$, parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.

math.CV↗

arXiv2020

Sufficient conditions for strong starlikeness

Let $p$ be an analytic function defined on the open unit disc $\mathbb{D}$ with $p(0)=1$ and $0< α\leq 1$. The conditions on complex valued functions $C$, $D$ and $E$ are obtained for $p$ to be subordinate to $((1+z)/(1-z))^α$ when $C(z) z^{2}p''(z)+D(z)zp'(z) + E(z)p(z)=0$. Sufficient conditions for confluent (Kummer) hypergeometric function and generalized and normalized Bessel function of the first kind of complex order to be subordinate to $((1+z)/(1-z))^α$ are obtained as applications. The conditions on $α$ and $β$ are derived for $p$ to be subordinate to $((1+z)/(1-z))^α$ when $1+βzp'(z)/p^{n}(z)$ with $n=1,2$ is subordinate to $1+4z/3+2z^{2}/3=:φ_{CAR}(z)$. Similar problems were investigated for $\RE p(z)>0$ when the functions $p(z)+βzp'(z)/p^{n}(z)$ with $n=0,2$ is subordinate to $φ_{CAR}(z)$. The condition on $β$ is determined for $p$ to be subordinate to $((1+z)/(1-z))^α$ when $p(z)+βzp'(z)/p^{n}(z)$ with $n=0,1,2$ is subordinate to $((1+z)/(1-z))^α$.

math.CV↗

arXiv2020

Starlikeness of certain analytic functions

Let $f$ and $g$ be analytic functions on the open unit disk of the complex plane with $f/g$ belonging to the class $\mathcal{P} $ of functions with positive real part consisting of functions $p$ with $p(0)=1$ and $\operatorname{Re} p(z)>0$ or to its subclass consisting of functions $p$ with $|p(z)-1|<1$. We obtain the sharp radius constants for the function $f$ to be starlike of order $α$, parabolic starlike, etc. when $g/k\in\mathcal{P}$ where $k$ denotes the Koebe function defined by $k(z)=z/(1-z)^2$.

math.CV↗

arXiv2020

Estimates for initial coefficients of certain bi-univalent functions

Estimates are obtained for the initial coefficients of a normalized analytic function $f$ in the unit disk $\mathbb{D}$ such that $f$ and the analytic extension of $f^{-1}$ to $\mathbb{D}$ belong to certain subclasses of univalent functions. The bounds obtained improve some existing known bounds.

math.CV↗