arXiv subjects

V. Ravichandran

Publications and source records attributed to V. Ravichandran.

At least 37 records · Page 2Linked to original sources

arXiv2020

Radius of Starlikeness for Classes of Analytic Functions

We consider normalized analytic function $f$ on the open unit disk for which either $\operatorname{Re} f(z)/g(z)>0$, $|f(z) /g(z) - 1|<1$ or $\operatorname{Re} (1-z^2) f(z) /z>0$ for some analytic function $g$ with $\operatorname{Re} (1-z^2) g(z) /z>0$. We have obtained the radii for these functions to belong to various subclasses of starlike functions. The subclasses considered include the classes of starlike functions of order $α$, lemniscate starlike functions and parabolic starlike functions.

math.CV↗

arXiv2020

Starlikeness for Certain Close-to-Star Functions

We find the radius of starlikeness of order $α$, $0\leq α<1$, of normalized analytic functions $f$ on the unit disk satisfying either $\operatorname{Re}(f(z)/g(z))>0$ or $\left| (f(z)/g(z))-1\right|<1$ for some close-to-star function $g$ with $\operatorname{Re}(g(z)/(z+z^2/2))>0$ as well as of the class of close-to-star functions $f$ satisfying $\operatorname{Re}(f(z)/(z+z^2/2))>0$. Several other radii such as radius of univalence and parabolic starlikeness are shown to be the same as the radius of starlikeness of appropriate order.

math.CV↗

arXiv2020

Radius of Starlikeness of Certain Analytic Functions

This paper studies analytic functions $f$ defined on the open unit disk of the complex plane for which $f/g$ and $(1+z)g/z$ are both functions with positive real part for some analytic function $g$. We determine radius constants of these functions to belong to classes of strong starlike functions, starlike functions of order $α$, parabolic starlike functions, as well as to the classes of starlike functions associated with lemniscate of Bernoulli, cardioid, lune, reverse lemniscate, sine function, exponential function and a particular rational function. The results obtained are sharp.

math.CV↗

arXiv2019

Coefficients of the Inverse Functions and Radius Estimates of Certain Starlike Functions

Ma-Minda class (of starlike functions) consists of all normalized analytic functions $f$ on the unit disk for which the image of $zf'(z)/f(z)$ is contained in the some starlike region in the right-half plane. We obtain the best possible bounds on the second and third coefficient for the inverse functions of functions in the Ma-Minda class. The bounds on the Fekete-Szegö functional and the second Hankel determinant of the inverse functions of the functions belonging to the Ma-Minda class are also determined. Further, the bounds on the first five coefficients of the inverse functions are investigated for two particular subclasses of the Ma-Minda class. In addition, some radius estimates associated with the two subclasses are also computed.

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

Radii of Starlikeness and Convexity of Bessel Functions

The radii of starlikeness and convexity associated with lemniscate of Bernoulli and the Janowski function, $(1+Az)/(1+Bz)$ for $-1\leq B<A\leq 1$, have been determined for normalizations of $q$-Bessel function, Bessel function of first kind of order $ν$, Lommel function of first kind and Legendre polynomial of odd degree.

math.CV↗

arXiv2019

Differential Subordination for Janowski Functions with Positive Real Part

Theory of differential subordination provides techniques to reduce differential subordination problems into verifying some simple algebraic condition called admissibility condition. We exploit the first order differential subordination theory to get several sufficient conditions for function satisfying several differential subordinations to be a Janowski function with positive real part. As applications, we obtain sufficient conditions for normalized analytic functions to be Janowski starlike functions.

math.CV↗

arXiv2019

Lemniscate Convexity and Other Properties of Generalized Bessel Functions

Sufficient conditions on associated parameters $p,b$ and $c$ are obtained so that the generalized and \textquotedblleft{normalized}\textquotedblright{} Bessel function $u_p(z)=u_{p,b,c}(z)$ satisfies $|(1+(zu''_p(z)/u'_p(z)))^2-1|<1$ or $|((zu_p(z))'/u_p(z))^2-1|<1$. We also determine the condition on these parameters so that $-(4(p+(b+1)/2)/c)u'_p(z)\prec\sqrt{1+z}$. Relations between the parameters $μ$ and $p$ are obtained such that the normalized Lommel function of first kind $h_{μ,p}(z)$ satisfies the subordination $1+(zh''_{μ,p}(z)/h'_{μ,p}(z))\prec\sqrt{1+z}$. Moreover, the properties of Alexander transform of the function $h_{μ,p}(z) $ are discussed.

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↗

arXiv2018

Harmonic Univalent Functions Defined by Post Quantum Calculus Operators

We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

math.CV↗

arXiv2018

Starlikeness associated with lemniscate of Bernoulli

For an analytic function $f$ on the unit disk $\mathbb{D}=\{z:|z|<1\}$ satisfying $f(0)=0=f'(0)-1,$ we obtain sufficient conditions so that $f$ satisfies $|(zf'(z)/f(z))^2-1|<1.$ The technique of differential subordination of first or second order is used. The admissibility conditions for lemniscate of Bernoulli are derived and employed in order to prove the main results.

math.CV↗

arXiv2017

Estimates for Coefficients of Certain Analytic Functions

For $ -1 \leq B \leq 1$ and $A>B$, let $\mathcal{S}^*[A,B]$ denote the class of generalized Janowski starlike functions consisting of all normalized analytic functions $f$ defined by the subordination $z f'(z)/f(z) \prec (1+ A z)/(1+ B z)$ $(|z|<1)$. For $-1 \leq B \leq 1 1)$ and $B=1$. As an application, for $F:=f^{-1}$, $A= 2 β-1$ $(β>1)$ and $B=1$, the sharp coefficient bounds of $F/F'$ are obtained when $f$ is a generalized Janowski starlike or generalized Janowski convex function. Further, we provide the sharp coefficient estimates for inverse functions of normalized analytic functions $f$ satisfying $f'(z) \prec (1+z)/(1+B z)$ $(|z|<1, -1 \leq B < 1)$.

math.CV↗

arXiv2017

First order differential subordination for functions with positive real part

Sharp estimates on $β$ are determined so that an analytic function $p$ defined on the open unit disk in the complex plane normalized by $p(0)=1$ is subordinate to some well known starlike functions with positive real part whenever $1+βz p'(z), \,\,1+βz p'(z)/p(z), \,\,\mbox{or}\,\,1+βz p'(z)/p^{2}(z)$ is subordinate to $\sqrt{1+z}$. Our results provide sharp version of previously known results.

math.CV↗

arXiv2017

Directional convexity of harmonic mappings

The convolution properties are discussed for the complex-valued harmonic functions in the unit disk $\mathbb{D}$ constructed from the harmonic shearing of the analytic function $ϕ(z):=\int_0^z (1/(1-2ξ\textit{e}^{\textit{i}μ}\cosν+ξ^2\textit{e}^{2\textit{i}μ}))\textit{d}ξ$, where $μ$ and $ν$ are real numbers. For any real number $α$ and harmonic function $f=h+\overline{g}$, define an analytic function $f_α:=h+\textit{e}^{-2\textit{i}α}g$. Let $μ_1$ and $μ_2$ $(μ_1+μ_2=μ)$ be real numbers, and $f=h+\overline{g}$ and $F=H+\overline{G}$ be locally-univalent and sense-preserving harmonic functions such that $f_{μ_1}*F_{μ_2}=ϕ$. It is shown that the convolution $f*F$ is univalent and convex in the direction of $-μ$, provided it is locally univalent and sense-preserving. Also, local-univalence of the above convolution $f*F$ is shown for some specific analytic dilatations of $f$ and $F$. Furthermore, if $g\equiv0$ and both the analytic functions $f_{μ_1}$ and $F_{μ_2}$ are convex, then the convolution $f*F$ is shown to be convex. These results extends the work done by Dorff \textit{et al.} to a larger class of functions.

math.CV↗

arXiv2017

Convolution of a harmonic mapping with $n$-starlike mappings and its partial sums

We investigate the univalency and the directional convexity of the convolution $ϕ\tilde{*}f=ϕ*h+\overline{ϕ*g}$ of the harmonic mapping $f=h+\bar{g}$ with a mapping $ϕ$ whose convolution with the mapping $z+\sum_{k=2}^{\infty}k^nz^k$ is starlike (and such a mapping $ϕ$ is called $n$-starlike). In addition, we investigate the directional convexity of (i) the convolution of an analytic convex mapping with the slanted half-plane mapping, and (ii) the partial sums of the convolution of a $6$-starlike mapping with the harmonic Koebe mapping and the harmonic half-plane mapping.

math.CV↗

arXiv2017

On Coefficient problem for bi-univalent analytic functions

Estimates for initial coefficients of Taylor-Maclaurin series of bi-univalent functions belonging to certain classes defined by subordination are obtained. Our estimates improve upon the earlier known estimates for second and third coefficient. The bound for the fourth coefficient is new. In addition, bound for the fifth coefficient is obtained for bi-starlike and strongly bi-starlike functions of order $ρ$ and $β$ respectively.

math.CV↗

arXiv2017

Convexity in one direction of convolutions and linear combination of harmonic functions

We show that the convolution of the harmonic function $f=h+\bar{g}$, where $h(z)+{e}^{-2{i}γ}g(z)=z/(1-{e}^{{i}γ}z)$ having analytic dilatation ${e}^{{i}θ} z^n (0\leqθ<2π)$, with the mapping $f_{a,α}=h_{a,α}+\overline{g}_{a,α}$, where $h_{a,α}(z)=(z/(1+a)-{e}^{{i}α}z^2/2)/(1-{e}^{{i}α}z)^2$, $g_{a,α}(z)=(a {e}^{2{i}α}z/(1+a)-{e}^{3{i}α}z^2/2)/(1-{e}^{{i}α}z)^2$ is convex in the direction $-(α+γ)$. We also show that the convolution of $f_{a,α}$ with the right half-plane mapping having dilatation $(a-z^2)/(1-az^2)$ is convex in the direction $-α$. Finally, we introduce a family of univalent harmonic mappings and find out sufficient conditions for convexity along imaginary-axis of the linear combinations of harmonic functions of this family.

math.CV↗

arXiv2016

Generalized Zalcman conjecture for some classes of analytic functions

For functions $f(z)= z+ a_2 z^2 + a_3 z^3 + \cdots$ in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional $ϕ(f,n,m;λ):=|λa_n a_m -a_{n+m-1}|$. For all real parameters $λ$ and $ β<1$, we provide the sharp upper bound of $ϕ(f,n,m;λ)$ for functions $f$ satisfying $\operatorname{Re}f'(z) > β$ and hence settles the open problem of estimating $ϕ(f,n,m;λ)$ recently proposed by Agrawal and Sahoo [S. Agrawal and S. k. Sahoo, On coefficient functionals associated with the Zalcman conjecture, arXiv preprint, 2016]. It is worth mentioning that the sharp estimations of $ϕ(f,n,m;λ)$ follow for starlike and convex functions of order $α$ $(α<1)$ when $λ\leq 0$. Moreover, for certain positive $λ$, the sharp estimation of $ϕ(f,n,m;λ)$ is given when $f$ is a typically real function or a univalent function with real coefficients or is in some subclasses of close-to-convex functions.

math.CV↗