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Saminathan Ponnusamy

Publications and source records attributed to Saminathan Ponnusamy.

At least 109 records · Page 6Linked to original sources
arXiv2016

Absolutely Convex, Uniformly Starlike and Uniformly Convex Harmonic Mappings

In this paper, we prove necessary and sufficient conditions for a sense-preserving harmonic function to be absolutely convex in the open unit disk. We also estimate the coefficient bound and obtain growth, covering and area theorems for absolutely convex harmonic mappings. A natural generalization of the classical Bernardi type operator for harmonic functions is considered and its connection between certain classes of uniformly starlike harmonic functions and uniformly convex harmonic functions is also investigated. At the end, as applications, we present a number of results connected with hypergeometric and polylogarithm functions.

math.CV↗
arXiv2016

Radii of covering disks for locally univalent harmonic mappings

For a univalent smooth mapping $f$ of the unit disk $\ID$ of complex plane onto the manifold $f(\ID)$, let $d_f(z_0)$ be the radius of the largest univalent disk on the manifold $f(\ID)$ centered at $f(z_0)$ ($|z_0|<1$). The main aim of the present article is to investigate how the radius $d_h(z_0)$ varies when the analytic function $h$ is replaced by a sense-preserving harmonic function $f=h+\overline{g}$. The main result includes sharp upper and lower bounds for the quotient $d_f(z_0)/d_h(z_0)$, especially, for a family of locally univalent $Q$-quasiconformal harmonic mappings $f=h+\overline{g}$ on $|z|<1$. In addition, estimate on the radius of the disk of convexity of functions belonging to certain linear invariant families of locally univalent $Q$-quasiconformal harmonic mappings of order $α$ is obtained.

math.CV↗
arXiv2016

On the generalized Zalcman functional $λa_n^2-a_{2n-1}$ in the close-to-convex family

Let ${\mathcal S}$ denote the class of all functions $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$ analytic and univalent in the unit disk $\ID$. For $f\in {\mathcal S}$, Zalcman conjectured that $|a_n^2-a_{2n-1}|\leq (n-1)^2$ for $n\geq 3$. This conjecture has been verified only certain values of $n$ for $f\in {\mathcal S}$ and for all $n\ge 4$ for the class $\mathcal C$ of close-to-convex functions (and also for a couple of other classes). In this paper we provide bounds of the generalized Zalcman coefficient functional $|λa_n^2-a_{2n-1}|$ for functions in $\mathcal C$ and for all $n\ge 3$, where $λ$ is a positive constant. In particular, our special case settles the open problem on the Zalcman inequality for $f\in \mathcal C$ (i.e. for the case $λ=1$ and $n=3$).

math.CV↗
arXiv2016

Generalized Zalcman conjecture for convex functions of order $α$

Let $\mathcal S$ denote the class of all functions of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$ which are analytic and univalent in the open unit disk $\ID$ and, for $λ>0$, let $Φ_λ(n,f)=λa_n^2-a_{2n-1}$ denote the generalized Zalcman coefficient functional. Zalcman conjectured that if $f\in \mathcal S$, then $|Φ_1 (n,f)|\leq (n-1)^2$ for $n\ge 3$. The functional of the form $Φ_λ(n,f)$ is indeed related to Fekete-Szegő functional of the $n$-th root transform of the corresponding function in $\mathcal S$. This conjecture has been verified for a certain special geometric subclasses of $\mathcal S$ but the conjecture remains open for $f\in {\mathcal S}$ and for $n > 6$. In the present paper, we prove sharp bounds on $|Φ_λ(n,f)|$ for $f\in \mathcal{F}(α)$ and for all $n\geq 3$, in the case that $λ$ is a positive real parameter, where $ \mathcal{F}(α)$ denotes the family of all functions $f\in {\mathcal S}$ satisfying the condition $${\rm Re } \left( 1+\frac{zf''(z)}{f'(z)}\right) > α~\mbox{ for $z\in \ID$}, $$ where $-1/2\leq α<1$. Thus, the present article proves the generalized Zalcman conjecture for convex functions of order $α$, $α\in [-1/2,1)$.

math.CV↗
arXiv2015

John disk and $K$-quasiconformal harmonic mappings

The main aim of this article is to establish certain relationships between $K$-quasiconformal harmonic mappings and John disks. The results of this article are the generalizations of the corresponding results of Ch.~Pommerenke \cite{Po}.

math.CV↗
arXiv2015

Bounds for the product of modified Bessel functions

In this note our aim is to present some monotonicity properties of the product of modified Bessel functions of first and second kind. Certain bounds for the product of modified Bessel functions of first and second kind are also obtained. These bounds improve and extend known bounds for the product of modified Bessel functions of first and second kind of order zero. A new Turán type inequality is also given for the product of modified Bessel functions, and some open problems are stated, which may be of interest for further research.

math.CA↗
arXiv2015

Geometric studies on the class ${\mathcal U}(λ)$

The article deals with the family ${\mathcal U}(λ)$ of all functions $f$ normalized and analytic in the unit disk such that $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<λ$ for some $0<λ\leq 1$. The family ${\mathcal U}(λ)$ has been studied extensively in the recent past and functions in this family are known to be univalent in $\ID$. However, the problem of determining sharp bounds for the second coefficients of functions in this family was solved recently in \cite{VY2013} by Vasudevarao and Yanagihara but the proof was complicated. In this article, we first present a simpler proof. We obtain a number of new subordination results for this family and their consequences. In addition, we show that the family ${\mathcal U}(λ)$ is preserved under a number of elementary transformations such as rotation, conjugation, dilation and omitted value transformations, but surprisingly this family is not preserved under the $n$-th root transformation for any $n\geq 2$. This is a basic here which helps to generate a number of new theorems and in particular provides a way for constructions of functions from the family ${\mathcal U}(λ)$. Finally, we deal with a radius problem.

math.CV↗
arXiv2015

Circular Symmetrization, Subordination and Arclength problems on Convex Functions

We study the class ${\mathcal C}(Ω)$ of univalent analytic functions $f$ in the unit disk $\mathbb{D} = \{z \in \mathbb{C} :\,|z|<1 \}$ of the form $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$ satisfying \[ 1+\frac{zf"(z)}{f'(z)} \in Ω, \quad z\in \mathbb{D}, \] where $Ω$ will be a proper subdomain of ${\mathbb C}$ which is starlike with respect to $1 (\in Ω)$. Let $ϕ_Ω$ be the unique conformal mapping of ${\mathbb D}$ onto $Ω$ with $ϕ_Ω(0)=1$ and $ϕ_Ω'(0) > 0$ and $ k_Ω(z) = \int_0^z \exp \left(\int_0^t ζ^{-1} (ϕ_Ω(ζ) -1) \, d ζ\right) \, dt$. Let $L_r(f)$ denote the arclength of the image of the circle $\{z \in \mathbb{C} : \, |z|=r\}$, $r\in (0,1)$. The first result in this paper is an inequality $L_r(f) \leq L_r(k_Ω)$ for $f \in \mathcal{C} (Ω)$, which solves the general extremal problem $\max_{f \in {\mathcal C}(Ω)} L_r(f)$, and contains many other well-known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class ${\mathcal C}(Ω)$.

math.CV↗
arXiv2015

Univalent harmonic mappings and lift to the minimal surfaces

We construct sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction, but with varying dilatation. Also, we obtain minimal surfaces associated with such harmonic mappings. This solves also a recent problem of Dorff and Muir (Abstr. Appl. Anal. (2014)). In several of the cases, we illustrate mappings together with their minimal surfaces pictorially with the help of \texttt{Mathematica} software.

math.CV↗
arXiv2015

Cross-product of Bessel functions: monotonicity patterns and functional inequalities

In this paper we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.

math.CA↗
arXiv2015

Injectivity of sections of convex harmonic mappings and convolution theorems

In the article the authors consider the class ${\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal{P}_H^0(α)$ and $\mathcal{G}_H^0(β)$ of functions from ${\mathcal H}_0$ and show that if $f\in \mathcal{P}_H^0(α)$ and $F\in\mathcal{G}_H^0(β)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $α$ and $β$ are satisfied. In the second part we study the harmonic sections (partial sums) $$ s_{n, n}(f)(z)=s_n(h)(z)+\overline{s_n(g)(z)}, $$ where $f=h+\overline{g}\in {\mathcal H}_0$, $s_n(h)$ and $s_n(g)$ denote the $n$-th partial sums of $h$ and $g$, respectively. We prove, among others, that if $f=h+\overline{g}\in{\mathcal H}_0$ is a univalent harmonic convex mapping, then $s_{n, n}(f)$ is univalent and close-to-convex in the disk $|z|< 1/4$ for $n\geq 2$, and $s_{n, n}(f)$ is also convex in the disk $|z|< 1/4$ for $n\geq2$ and $n\neq 3$. Moreover, we show that the section $s_{3,3}(f)$ of $f\in {\mathcal C}_H^0$ is not convex in the disk $|z|<1/4$ but is shown to be convex in a smaller disk.

math.CV↗
arXiv2015

Where is $f(z)/f'(z)$ univalent?

Let ${\mathcal S}$ denote the family of all univalent functions $f$ in the unit disk $\ID$ with the normalization $f(0)=0= f'(0)-1$. There is an intimate relationship between the operator $P_f(z)=f(z)/f'(z)$ and the Danikas-Ruscheweyh operator $T_f:=\int_{0}^{z}(tf'(t)/f(t))\,dt$. In this paper we mainly consider the univalence problem of $F=P_f$, where $f$ belongs to some subclasses of ${\mathcal S}$. Among several sharp results and non-sharp results, we also show that if $f\in {\mathcal S}$, then $F \in {\mathcal U}$ in the disk $|z|<r$ with $r\leq r_6\approx 0.360794$ and conjecture that the upper bound for such $r$ is $\sqrt{2}-1$.

math.CV↗
arXiv2015

Turán type inequalities for general Bessel functions

In this paper some Turán type inequalities for the general Bessel function, monotonicity and bounds for its logarithmic derivative are derived. Moreover we find the series representation and the relative extrema of the Turánian of general Bessel functions. The key tools in the proofs are the recurrence relations together with some asymptotic relations for Bessel functions.

math.CA↗
arXiv2014

Turán type inequalities for confluent hypergeometric functions of the second kind

In this paper we deduce some tight Turán type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature. We also give alternative proofs for some already established Turán type inequalities. Moreover, by using these Turán type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometric functions of the second kind. The key tool in the proof of the Turán type inequalities is an integral representation for a quotient of Tricomi confluent hypergeometric functions, which arises in the study of the infinite divisibility of the Fisher-Snedecor $F$ distribution.

math.CA↗
arXiv2014

Maximal area integral problem for certain class of univalent analytic functions

One of the classical problems concerns the class of analytic functions $f$ on the open unit disk $|z|<1$ which have finite Dirichlet integral $Δ(1,f)$, where $$Δ(r,f)=\iint_{|z|<r}|f'(z)|^2 \, dxdy \quad (0<r\leq 1). $$ The class ${\mathcal S}^*(A,B)$ of normalized functions $f$ analytic in $|z|<1$ and satisfies the subordination condition $zf'(z)/f(z)\prec (1+Az)/(1+Bz)$ in $|z|<1$ and for some $-1\leq B\leq 0$, $A\in {\mathbb C}$ with $A\neq B$, has been studied extensively. In this paper, we solve the extremal problem of determining the value of $$\max_{f\in {\mathcal S}^*(A,B)}Δ(r,z/f)$$ as a function of $r$. This settles the question raised by Ponnusamy and Wirths in [11]. One of the particular cases includes solution to a conjecture of Yamashita which was settled recently by Obradović et. al [9].

math.CV↗
arXiv2014

Turán type inequalities for Struve functions

Some Turán type inequalities for Struve functions of the first kind are deduced by using various methods developed in the case of Bessel functions of the first and second kind. New formulas, like Mittag-Leffler expansion, infinite product representation for Struve functions of the first kind, are obtained, which may be of independent interest. Moreover, some complete monotonicity results and functional inequalities are deduced for Struve functions of the second kind. These results complement naturally the known results for a particular case of Lommel functions of the first kind, and for modified Struve functions of the first and second kind.

math.CA↗
arXiv2013

On characterizations of Bloch-type, Hardy-type and Lipschitz-type spaces

In this paper, we establish a Bloch-type growth theorem for generalized Bloch-type spaces and discuss relationships between Dirichlet-type spaces and Hardy-type spaces on certain classes of complex-valued functions. Then we present some applications to non-homogeneous Yukawa PDEs. We also consider some properties of the Lipschitz-type spaces on certain classes of complex-valued functions. Finally, we will study a class of composition operators on these spaces.

math.CV↗