On starlikeness and convexity of certain analytic functions

Lemma 2.2. Let $w_1(z) = [1 - w(z)]/[1 + \beta w(z)]$ , where $w(z) \in \mathscr{B}$ , $\beta \geq 0$ . Then, fo $|z| = r < \min(1, 1/\beta)$ , $$\begin{split} \text{Re} \left\{ -\beta w_{_1}(z) + \frac{1}{w_{_1}(z)} \right\} + \frac{r^2 |1 + \beta w_{_1}(z)|^2 - |1 - w_{_1}(z)|^2}{(1 - r^2) |w_{_1}(z)|} \ & \leq \frac{1 - \beta + (3\beta + 1)r + \beta (\beta + 3)r^2 + \beta (\beta - 1)r^3}{(1 - r^2)(1 + \beta r)} \,. \end{split}$$

Lemma 2.3. If $w(z) \in \mathscr{B}$ , $\beta \geq 0$ , then for $|z| = r < \min(1, 1/\beta)$ , (2.3) $$\operatorname{Re}\left\{\frac{zw'(z)}{[1-w(z)][1+\beta w(z)]}\right\} \leq \frac{r}{(1-r)(1+\beta r)}$$ . Proof. From Lemma 2.1, we have $$\operatorname{Re}\left\{\frac{zw'(z)}{(1-w(z))(1+\beta w(z))}\right\} \le \operatorname{Re}\left\{\frac{w(z)}{(1-w(z))(1+\beta w(z))}\right\} \ + \frac{r^2-|w(z)|^2}{(1-r^2)|1-w(z)||1+\beta w(z)|} \,.$$ Put $w_i(z) = [1 - w(z)]/[1 + \beta w(z)]$ , then the above inequality becomes $$\begin{split} \text{Re} \left\{ \frac{zw'(z)}{(1-w(z))(1+\beta w(z))} \right\} & \leq \frac{1}{(1+\beta)^2} \bigg[\beta - 1 + \text{Re} \left\{ -\beta w_{_1}\!(z) + \frac{1}{w_{_1}\!(z)} \right\} \ & + \frac{r^2 |1+\beta w_{_1}\!(z)|^2 - |1-w_{_1}\!(z)|^2}{(1-r^2)|w_{_1}\!(z)|} \bigg] \,. \end{split}$$ An application of Lemma 2.2 to the right hand side will give the result which is easily seen to be sharp for w(z) = z at z = r. The following lemma is a consequence of [2, Theorem 3].

Lemma 2.4. If $p(z) \in \mathcal{P}$ , then on |z| = r, (2.4) $$\operatorname{Re}\left\{\frac{zp'(z)}{1+p(z)}\right\} \ge \begin{cases} -\frac{r}{1+r} & , \quad for \quad r < \frac{1}{3} \\ \frac{r^2+2^{3/2}(1-r^2)^{1/2}-3}{1-r^2} & , \quad for \quad \frac{1}{3} \le r < 1 \end{cases}.$$ (2.5) $$\operatorname{Re}\left\{\frac{zp'(z)}{n(z)}\right\} \ge -\frac{2r}{1-r^2}.$$

Theorem 3.1. Let $f(z) \in N$ be such that $f(z)/[\lambda f(z) + (1-\lambda)g(z)] \in \mathbb{Z}_7$ , where $g(z) \in N$ and $g(z)/z \in \mathbb{Z}_7$ , $0 \le \lambda < (1+\sqrt{3}+1/2\gamma)/(2+\sqrt{3})$ . Then the radius of starlikeness $\sigma_1$ of f(z) is given by the only positive root in (0,1) of the equation $$\beta r^3 + (2+3\beta)r^2 + 3r - 1 = 0$$ , where $\beta = [(1 + \lambda)\gamma - 1]/(1 - \lambda)\gamma$ .

Theorem 3.2. Let $f(z) \in N$ be such that $f(z)/[\lambda f(z) + (1-\lambda)g(z)] \in \mathscr{Q}_{\tau}$ , where $g(z) \in N$ and $g(z)/z \in \mathscr{S}_{1/2}$ . Then the radius of starlikeness of f(z) is $$\sigma_{\scriptscriptstyle 2} = \begin{cases} r_{\scriptscriptstyle 1} \ , & for \ 0 \leqq \lambda \leqq 1/2 \gamma \ , \ [2^{\scriptscriptstyle 1/2}(1+\beta)^{\scriptscriptstyle 1/2} - 1]/(1+2\beta) \ , & for \ 1/2 \gamma < \lambda < (\sqrt{5} + 1 \ + 1/\gamma)/(\sqrt{5} + 3) \ , \end{cases}$$ where $\beta = [(1 + \lambda)\gamma - 1]/(1 - \lambda)\gamma$ and $r_1$ is the smallest positive root in (0, 1) of the equation $$\begin{aligned} (1+2\beta+9\beta^2)r^4+2(1+12\beta+3\beta^2)r^3+(13+10\beta+\beta^2)r^2\ +4(1-\beta)r-4=0 \;. \end{aligned}$$

Theorem 3.3. Let $f(z) \in N$ be such that $f(z)/[\lambda f(z) + (1-\lambda)g(z)] \in \mathscr{Q}_7$ , where $g(z) \in S^*(\alpha)$ , $0 \le \lambda < \lambda_0$ , some $\lambda_0 < 1$ . Then the radius of starlikeness $\sigma_3$ of f(z) is given by the smallest positive root in (0, 1) of the equation $$\beta(2\alpha-1)r^3+(3\beta+2\alpha-2\alpha\beta)r^2+(3-2\alpha)r-1=0$$ , where $\beta = [(1 + \lambda)\gamma - 1]/(1 - \lambda)\gamma$ .