New subclasses of Ozaka's convex functions

Document Type : Research Paper


Department of Mathematics, Faculty of Sciences, Urmia University, P. O. Box 165, Urmia, Iran


Let $\mathcal{S}^{\ast}_{L}(\uplambda)$ and $\mathcal{CV}_L(\uplambda)$ be the classes of functions
$f$, analytic in the unit disc $\Updelta=\{z\colon|z|<1\}$, with the
normalization $f(0)=f'(0)-1=0$, which satisfies the conditions
\frac{zf'(z)}{f(z)}\prec \left(1+z\right)^{\uplambda}\quad\text{and}\quad \left(1+\frac{zf''(z)}{f'(z)}\right)\prec \left(1+z\right)^{\uplambda}
\qquad \left(0<\uplambda\le 1 \right),
where $\prec$ is the subordination relation, respectively. The classes
$\mathcal{S}^{\ast}_{L}(\uplambda)$ and $\mathcal{CV}_L(\uplambda)$ are subfamilies of the known classes of strongly starlike and convex functions of order $\uplambda$. We consider  the relations between $\mathcal{S}^{\ast}_{L}(\uplambda)$, $\mathcal{CV}_L(\uplambda)$ and other classes geometrically defined. Also, we obtain the sharp radius of convexity for functions belonging to $\mathcal{S}^{\ast}_{L}(\uplambda)$ class. Furthermore,  the norm of pre-Schwarzian derivatives  and univalency of functions $f$ which satisfy the condition
\myp{z \in \Updelta},
are considered.


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Volume 14, Issue 3
March 2023
Pages 189-199
  • Receive Date: 18 July 2022
  • Revise Date: 30 May 2022
  • Accept Date: 12 October 2022