Regular functions $f(z)$ for which $z f’(z)$ is $\alpha$-spiral

Abstract:

A function $f(z) = z + \Sigma _{n = 2}^\infty {a_n}{z^n}$ regular in the open unit disk $\Delta = \{ z:|z| < 1\}$ is a (univalent) $\alpha$-spiral function for real $\alpha ,|\alpha | < \pi /2$, if $\operatorname {Re} \{ {e^{i\alpha }}zf’(z)/f(z)\} > 0$ for z in $\Delta$; in this case we write $f(z) \in {\mathcal {F}_\alpha }$. A fundamental result of this paper shows that the transformation \[ {f_ \ast }(z) = \frac {{azf((z + a)/(1 + \bar az))}}{{f(a)(z + a){{(1 + \bar az)}^{{e^{ - 2i\alpha }}}}}}\] defines a function in ${\mathcal {F}_\alpha }$ whenever $f(z)$ is in ${\mathcal {F}_\alpha }$ and a is in $\Delta$. If $g(z)$ is regular in $\Delta ,g(0) = 0$ and $g’(0) = 1$, then $g(z)$ is in ${\mathcal {G}_\alpha }$ if and only if $zg’(z)$ is in ${\mathcal {F}_\alpha }$. The main result of the paper is the derivation of the sharp radius of close-to-convexity for each class ${\mathcal {G}_\alpha }$; it is given as the solution of an equation in r which is dependent only on $\alpha$. (Approximate solutions of this equation were made by computer and these suggest that the radius of close-to-convexity of the class $\mathcal {G} = { \cup _\alpha }{\mathcal {G}_\alpha }$ is approximately $.99097^{+}$.) Additional results are also obtained such as the radius of convexity of ${\mathcal {G}_\alpha }$, a range of $\alpha$ for which $g(z)$ in ${\mathcal {G}_\alpha }$ is always univalent is given, etc. These conclusions all depend heavily on the transformation cited above and its analogue for ${\mathcal {G}_\alpha }$.