arXiv subjects
Publications and source records attributed to Priyanka Goel.
In the present investigation, we study the class of Sigmoid starlike functions, given by $\mathcal{S}^*_{SG}=\{f\in\mathcal{A}: {zf'(z)}/{f(z)}\prec 2/(1+e^{-z})\}$ in context of estimating the sharp radius constants associated with several known subclasses of starlike functions. Further, graphical validation for the sharpness of results is also provided.
For $0\leq\alpha\leq 1,$ let $H_{\alpha}(x,y)$ be the convex weighted harmonic mean of $x$ and $y.$ We establish differential subordination implications of the form \begin{equation*} H_{\alpha}(p(z),p(z)\Theta(z)+zp'(z)\Phi(z))\prec h(z)\Rightarrow p(z)\prec h(z), \end{equation*} where $\Phi,\;\Theta$ are analytic functions and $h$ is a univalent function satisfying some special properties. Further, we prove differential subordination implications involving a combination of three classical means. As an application, we generalize many existing results and obtain sufficient conditions for starlikeness and univalence.
For $0<\alpha\leq1$ and $\lambda>0,$ let \begin{equation}\label{1} G_{\lambda,\alpha}=\left\{f\in\mathcal{A}:\left|\dfrac{1-\alpha+\alpha zf''(z)/f'(z)}{zf'(z)/f(z)}-(1-\alpha)\right|<\lambda, z\in\mathbb{D}\right\}, \end{equation} the general form of Silverman class introduced by Tuneski and Irnak. For this class we derive some sufficient conditions in the form of differential inequalities. Further, we consider the class $\Omega,$ given by \begin{equation}\label{omega} \Omega=\left\{f\in\mathcal{A}:|zf'(z)-f(z)|<\dfrac{1}{2},\;z\in\mathbb{D}\right\}. \end{equation} For the above two classes, we establish inclusion relations involving some other well known subclasses of $\mathcal{S}^*$ and find radius estimates for different pairs involving these classes.
We introduce and study the following special type of differential subordination implication: \begin{equation}\label{abs} p(z)Q(z)+\frac{zp'(z)}{\beta p(z)+\alpha}\prec h(z)\quad\Rightarrow p(z)\prec h(z), \end{equation} which generalizes the Briot-Bouquet differential subordination, where $Q(z)$ is analytic and $0\neq\beta,\alpha\in\mathbb{C}.$ Further, some special cases of our result are also discussed. Finally, analogues of open door lemma and integral existence theorem with applications to univalent functions are obtained.
We derive general formula for the fourth coefficient of the functions belonging to the Carathéodory class involving the parameters lying in the open unit disk. Further, we obtain sharp upper bounds of initial inverse coefficients for certain close-to-convex functions satisfying any one of the inequalities: $\RE((1-z)f'(z))>0,$ $\RE((1-z^2)f'(z))>0,$ $\RE((1-z+z^2)f'(z))>0$ and $\RE((1-z)^2f'(z))>0$.