Convex hulls and extreme points of families of starlike and close-to-convex mappings

Theorem 1. Let $X = \{x : |x| = 1 \text{ and } \text{Im } x \geq 0\}$ , $\alpha \geq 1$ , and $\mathscr{S}$ denote the set of probability measures on X. Given $v \in \mathscr{S}$ there exists a $\mu \in \mathscr{S}$ so that $$\left[\int_x \frac{1-z^2}{(1-xz)(1-\overline{x}z)} dv(x)\right]^{\alpha} = \int_x \left[\frac{1-z^2}{(1-xz)(1-\overline{x}z)}\right]^{\alpha} d\mu(x) .$$

Theorem 2. Let $$F_p = \left\{ \int_x \frac{1}{[(1 - xz)(1 - \overline{x}z)]^p} d\mu(x) \colon \mu \in \mathscr{S} \right\}$$ where X and $\mathscr F$ are as in Theorem 1. If p>0 and q>0 then $F_p\cdot F_q\subset F_{q+q}$ where $F_p\cdot F_q=[f\colon f=gh \ and \ g\in F_p,\ h\in F_q\}.$

Theorem 3. Let X and $\mathscr S$ be as in Theorem 1. Then given $\mu\in\mathscr S$ , $\exists v\in\mathscr S$ such that $$\exp\left\{\int_{x}-p\log(1-xz)(1-\bar{x}z)d\mu(x)\right\}=\int_{x}[(1-xz)(1-\bar{x}z)]^{-p}dv(x).$$

Theorem 4. Let X and $\mathscr S$ be as in Theorem 1, $\alpha<1$ , k be any positive integer and $\mathscr F$ be the set of functions $f_\mu$ on $\Delta$ defined by $$f_{u}(z) = \int_{x} \frac{z}{(1 - xz^{k})^{(1-\alpha)/k}(1 - \bar{x}z^{k})^{(1-\alpha)/k}} \qquad (\mu \in \mathscr{S})$$ then $\mathscr{F} = \mathscr{H}St_{R}(\alpha, k)$ and $$\mathscr{EHS}t_{\scriptscriptstyle N}(\alpha,\,k)\subset \left\{\frac{z}{(1-xz^k)^{(1-\alpha)/k}(1-\overline{x}z^k)^{(1-\alpha)/k}}\colon |x|=1,\, \mathrm{Im}\; x\geqq 0\right\}\;.$$

Lemma 1. Suppose Re p(z) > 0, p(0) = 1, p(z) is even, and p(z) is real on (-1, 1). $$p(z) = \int_{x} \frac{1 - z^{4}}{(1 - xz^{2})(1 - \bar{x}z^{2})} d\mu(x)$$ where $X = \{x : |x| = 1 \text{ and } \text{Im } x \ge 0\}$ and $\mu$ is a probability measure on X.

Theorem 5. Let $X = \{x : |x| = 1, \text{ Im } x \ge 0\}$ , $\mathscr P$ be the set of probability measures on X, and $\mathscr F$ be the set of functions $f_{\mu}$ on $\Delta$ defined by $$f_{\mu}(z) = \int_{x} \frac{z(1+z^2)}{(1-xz^2)(1-\overline{x}z^2)} d\mu(x) \quad (\mu \in \mathscr{P}) \; .$$ Let $T_2$ be the set of typically real odd functions on $\Delta$ . Then $\mathcal{H} T_2 = T_2 = \mathcal{F}$ , the map $\mu \to f_{\mu}$ is one-to-one, and each function $$z \longrightarrow \frac{z(1+z^2)}{(1-xz^2)(1-\overline{x}z^2)}$$ is an extreme point of $\mathcal{H}T_2$ .

Theorem 6. Let $\{X=x\colon |x|=1, \text{ Im } x\geq 0\}$ , $\mathscr P$ be the set of probability measures on X, and $\mathscr F$ be the class of functions $f_\mu$ on $\Delta$ defined by $$f_{\mu}(z) = \int_{x} \left[ \frac{1-z^2}{(1-xz)(1-\overline{x}z)} \right]^{\beta} \frac{1}{(1-xz)(1-\overline{x}z)} d\mu(x) \quad (\mu \in \mathscr{P}) \ .$$ Then $\mathscr{F} = \mathscr{H}K'_{\mathbb{R}}(\beta)$ , where $\beta \geq 1$ and $K'_{\mathbb{R}}(\beta)$ . Also $$\mathscr{CH}K'_R(\beta) \subset \left\{ \left[ \frac{1-z^2}{(1-xz)(1-\bar{x}z)} \right]^{\beta} \frac{1}{(1-xz)(1-\bar{x}z)} \colon |x| \right.$$ $=1, \ \mathrm{Im} \ x \geqq 0 \right\} \ .$

Theorem 7. Suppose $F(z) \in K(\beta)$ where $\beta \ge 1$ . If f(z) < F(z) then $f(z) \ll f_k(z)$ where $$f_k(z) = \frac{1}{k} \Big[ \Big(\frac{1+z}{1-z}\Big)^{k/2} - 1 \Big] \quad and \quad \beta = \frac{1}{2}k - 1 \; .$$

Theorem 8. Let $X^2 = \{(x, y): |x| = |y| = 1\}$ , $\mathscr T$ be the set of probability measures on $X^2$ , and $\mathscr F$ be the class of functions $f_{\mu}$ on $\Delta$ defined by $$f_{\mu}(z) = \int_{x^2} \frac{1 - xz^k}{(1 - yz^k)^{2/k+1}} d\mu(x, y) \quad (\mu \in \mathscr{P})$$ then $\mathscr{F} = \mathscr{H}C'_k$ , where $C'_k$ is the set of derivatives of functions in $C_k$ and $k = 1, 2, \cdots$ . Furthermore $$\mathscr{E}\mathscr{H}C_k'\subset\left\{\frac{1+xz^k}{(1-yz^k)^{2/k+1}};|x|=|y|=1\right\}$$ .