Criteria for univalence, starlikeness and convexity
Volume 85 / 2005
Abstract
Let ${\mathcal A}$ denote the class of all normalized analytic functions $f$ ($f(0)=0= f'(0)-1$) in the open unit disc $\mit\Delta$. For $0<\lambda\leq 1$, define $${\mathcal U}(\lambda )=\bigg \{f\in {\mathcal A}: \bigg|\bigg(\frac{z}{f(z)}\bigg)^{2}f'(z)-1\bigg|<\lambda, \, z\in {\mit\Delta} \bigg \} $$ and $$ {\mathcal P}(2\lambda )=\bigg \{f\in {\mathcal A}: \bigg|\bigg(\frac{z}{f(z)}\bigg)' '\bigg|<2\lambda, \, z\in {\mit\Delta}\bigg \}. $$ Recently, the problem of finding the starlikeness of these classes has been considered by Obradović and Ponnusamy, and later by Obradović et al. In this paper, the authors consider the problem of finding the order of starlikeness and of convexity of ${\mathcal U}(\lambda )$ and ${\mathcal P}(2\lambda )$, respectively. In particular, for $f\in {\mathcal A}$ with $f' '(0)=0$, we find conditions on $\lambda$, $\beta^* (\lambda )$ and $\beta (\lambda )$ so that ${\mathcal U}(\lambda ) \subsetneq {\mathcal S}^*(\beta^* (\lambda ))$ and ${\mathcal P}(2\lambda )\subsetneq {\mathcal K}(\beta (\lambda ))$. Here, ${\mathcal S}^*(\beta)$ and ${\mathcal K}(\beta)$ ($\beta <1$) denote the classes of functions in ${\mathcal A}$ that are starlike of order $\beta$ and convex of order $\beta$, respectively. In addition to these results, we also provide a coefficient condition for functions to be in ${\mathcal K}(\beta)$. Finally, we propose a conjecture that each function $f\in {\mathcal U}(\lambda )$ with $f' '(0)=0$ is convex at least when $0<\lambda\leq 3-2\sqrt{2}$.
