Sufficient conditions and radius problems for the Silverman class

  • S. Sivaprasad Kumar Department of Applied Mathematics, Delhi Technological University, India
  • Priyanka Priyanka Department of Applied Mathematics, Delhi Technological University, India
Keywords: Silverman class, starlike functions, radius problems


UDC 517.5

For $0<\alpha\leq1$ and $\lambda>0,$ let \begin{equation}\label{1} G_{\lambda,\alpha}=\left\{f\in\mathcal{A}\colon \left|\frac{1-\alpha+\alpha zf''(z)/f'(z)}{z f'(z)/f(z)}-(1-\alpha)\right|<\lambda,\ z\in\mathbb{D}\right\}.\tag{0.1}\end{equation} The general form of the Silverman class introduced by Tuneski and Irmak [Int. J. Math. and Math. Sci., {\bf 2006}, Article~ID 38089 (2006)].  Our differential inequality formulation lays out several sufficient conditions for this class.  Further, we consider a class $\Omega$ given by \begin{equation}\label{omega}\Omega=\left\{f\in\mathcal{A}\colon  |zf'(z)-f(z)|<\frac{1}{2},\ z\in\mathbb{D}\right\}.\tag{0.2}\end{equation} For these two classes, we establish inclusion relations involving some well-known subclasses of $\mathcal{S}^*$ and compute radius estimates featuring various pairings of these classes.


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How to Cite
Kumar, S. S., and P. Priyanka. “Sufficient Conditions and Radius Problems for the Silverman Class”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, no. 3, Mar. 2024, pp. 405 -22, doi:10.3842/umzh.v76i3.7317.
Research articles