Univalency of certain transform of univalent functions

Theorem 1. Let $f \in \mathcal{U}$ . Then the function g defined by the equation (4) belongs to $\mathcal{U}$ in the disc $$|z| < \sqrt{\frac{1 - |a_2| + \sqrt{|a_2|^2 + 2|a_2| - 3}}{2}},$$ i.e., satisfies (2) on this disc, if $\frac{5}{4} \leq |a_2| \leq 2$ .

Lemma 1. Let $f \in A$ be of the form (1). If (8) $$\sum_{n=0}^{\infty} n|a_n| \le 1,$$ then $$|f'(z) - 1| < 1 \qquad (z \in \mathbb{D}),$$ $$\left| \frac{zf'(z)}{f(z)} - 1 \right| < 1 \qquad (z \in \mathbb{D})$$ (i.e. $f \in \mathcal{S}^*$ ), and $f \in \mathcal{U}$ . For the proof of $f \in \mathcal{U}$ in the lemma see [4], while the rest easily follows. Further, let $\mathcal{S}^+$ denote the class of univalent functions in the unit disc with the representation (9) $$\frac{z}{f(z)} = 1 + b_1 z + b_2 z^2 + \dots, \quad b_n \ge 0, \ n = 1, 2, 3, \dots$$ For example, the Silverman class (the class with negative coefficients) is included in the class $S^+$ , as well as the Koebe function $k(z) = \frac{z}{(1+z)^2} \in S^+$ . The next characterization is valid for the class $S^+$ (for details see [3]) (10) $$f \in \mathcal{S}^+ \quad \Leftrightarrow \quad \sum_{n=2}^{\infty} (n-1)b_n \le 1.$$

Theorem 2. Let $f \in S^+$ . Then the function g defined by (4) belongs to the class $\mathcal{U}$ in the disc $|z| < |a_2|/2$ and the result is the best possible.

Theorem 3. Let $f \in S$ . Then the function g defined by (4) belongs to the class $\mathcal{U}$ in the disc $|z| < r_0$ , where $r_0$ is the unique real root of the equation (12) $$\frac{3r^2 - 2r^4}{(1 - r^2)^2} - \ln(1 - r^2) = |a_2|^2$$ on the interval (0,1).