arXiv subjects
Publications and source records attributed to A. Vasudevarao.
Let $\mathcal{S}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$ of the form $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$. In this paper, we determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in $\mathcal{S}$ and its certain subclasses. We also discuss similar problems for typically real functions.
In this article we determine the coefficient bounds for functions in certain subclasses of analytic functions defined by subordination which are related to the well-known classes of starlike and convex functions. The main results deal with some open problems proposed by Q.H. Xu et al. [20,21]. An application of Jack lemma for certain subclass of starlike functions has been discussed.
For an analytic and univalent function $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$, the logarithmic coefficients $γ_n$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n$. In the present paper, we consider the class of close-to-convex functions (with argument $0$), and determine the sharp upper bound of $|γ_3|$ for such functions $f$, which proves a recent conjecture of the first and third authors [1].
The logarithmic coefficients $γ_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n$. In the present paper, we consider close-to-convex functions (with argument $0$) with respect to odd starlike functions and determine the sharp upper bound of $|γ_n|$, $n=1,2,3$ for such functions $f$.
Let $\mathcal{S}$ denote the class of functions analytic and univalent (i.e. one-to-one) in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:\, |z|<1\}$ normalized by $f(0)=0=f'(0)-1$. The logarithmic coefficients $γ_n$ of $f\in\mathcal{S}$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n.$ In the present paper, we determine the sharp upper bounds for $|γ_1|$, $|γ_2|$ and $|γ_3|$ when $f$ belongs to some familiar subclasses of close-to-convex functions.
For $-1\le B 1$. For $f\in\mathcal{S}^*(A,B)$ and $λ>0$, we shall estimate the absolute value of the Taylor coefficients $a_n(-λ,f)$ of the analytic function $(f(z)/z)^{-λ}$. Using this we shall determine the coefficient estimate for inverses of functions in the classes $\mathcal{S}^*(A,B)$ and $Σ^*(A,B)$.
Let $\mathcal{H}$ denote the class of harmonic functions $f$ in $\mathbb{D}:= \{z\in \mathbb{C}:|z| < 1\}$ normalized by $f(0) = 0 = f_z(0) -1$. For $α\geq 0$, we consider the following class $$\mathcal{W}^0_{\mathcal{H}}(α):= \{f = h + \overline{g}\in\mathcal{H}: {\rm Re\,}(h'(z) + αz h''(z)) >|g'(z) + αz g''(z)|, \quad z\in \mathbb{D}\}. $$ In this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for functions in the class $\mathcal{W}^0_{\mathcal{H}}(α)$. We also prove growth theorem, convolution, convex combination properties for functions in the class $\mathcal{W}^0_{\mathcal{H}}(α)$. Finally, we determine the value of $r$ so that the partial sums of functions in the class $\mathcal{W}^0_{\mathcal{H}}(α)$ are close-to-convex in $|z|<r$.
The logarithmic coefficients $γ_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n$. Recently, D.K. Thomas [On the logarithmic coefficients of close to convex functions, {\it Proc. Amer. Math. Soc.} {\bf 144} (2016), 1681--1687] proved that $|γ_3|\le \frac{7}{12}$ for functions in a subclass of close-to-convex functions (with argument $0$) and claimed that the estimate is sharp by providing a form of a extremal function. In the present paper, we pointed out that such extremal functions do not exist and the estimate is not sharp by providing a much more improved bound for the whole class of close-to-convex functions (with argument $0$). We also determine a sharp upper bound of $|γ_3|$ for close-to-convex functions (with argument $0$) with respect to the Koebe function.
For $γ\in\IC$ such that $|γ|<π/2$ and $0\leqβ<1$, let ${\mathcal P}_{γ,β} $ denote the class of all analytic functions $P$ in the unit disk $\mathbb{D}$ with $P(0)=1$ and $$ {\rm Re\,} \left (e^{iγ}P(z)\right)>β\cosγ\quad \mbox{ in ${\mathbb D}$}. $$ For any fixed $z_0\in\mathbb{D}$ and $λ\in\overline{\mathbb{D}}$, we shall determine the region of variability $V_{\mathcal{P}}(z_0,λ)$ for $\int_0^{z_0}P(ζ)\,dζ$ when $P$ ranges over the class $$ \mathcal{P}(λ) = \left\{ P\in{\mathcal P}_{γ,β} :\, P'(0)=2(1-β)λe^{-iγ}\cosγ\right\}. $$ As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.
For $α\in\IC\setminus \{0\}$ let $\mathcal{E}(α)$ denote the class of all univalent functions $f$ in the unit disk $\mathbb{D}$ and is given by $f(z)=z+a_2z^2+a_3z^3+\cdots$, satisfying $$ {\rm Re\,} \left (1+ \frac{zf''(z)}{f'(z)}+αzf'(z)\right)>0 \quad {in ${\mathbb D}$}. $$ For any fixed $z_0$ in the unit disk $\mathbb{D}$ and $λ\in\overline{\mathbb{D}}$, we determine the region of variability $V(z_0,λ)$ for $\log f'(z_0)+αf(z_0)$ when $f$ ranges over the class $$\mathcal{F}_α(λ)=\left\{f\in\mathcal{E}(α) \colon f''(0)=2λ-α%\quad{and} f'''(0)=2[(1-|λ|^2)a+ %(λ-α)^2 -λα] \right\}. $$ We geometrically illustrate the region of variability $V(z_0,λ)$ for several sets of parameters using Mathematica. In the final section of this article we propose some open problems.
In this paper we determine the region of variability for spirallike funcions with respect to a boundary point. In the final section we graphically illustrate the region of variability for several sets of parameters.
In this paper we determine the region of variability for certain subclasses of univalent functions satisfying differential inequalities. In the final section we graphically illustrate the region of variability for several sets of parameters.
Multiferroicity can be induced in strontium titanate by applying biaxial strain, resulting in the coexistence of both ferroelectric and antiferrodistortive domains. The magnitude and sign of the strain imposed on the lattice by design can be used to tune the phase transitions and interactions between these two phenomena. Using optical second harmonic generation, we report a transition from centrosymmetric 4/mmm phase to ferroelectric mm2, followed by an antiferrodistortive transition to a coupled ferroelastic-ferroelectric mm2 phase in a strontium titanate thin film strained in biaxial tension by 0.94%. The results agree well with theoretical first principles and phase-field predictions. Direct imaging of domains arising from the ferroelectric phase transition, and its switching under electric fields is demonstrated using piezoelectric force microscopy. Nonlinear optics combined with phase-field modeling is used to show that the dominant multiferroic domain switching mechanism is through coupled 90 degree ferroelectric-ferroelastic domain wall motion. More broadly, these studies of coexisting ferroelectric (polar) and antiferrodistortive rotation (axial) phenomena could have relevance to multiferroics with coexisting ferroelectric (polar) and magnetic (axial) phenomena.