Some inequalities for a certain subclass of starlike functions

Lemma 1.1. [10, p.35] Let $\Xi$ be a set in the complex plane $\mathbb{C}$ and let b be a complex number such that Re(b) > 0. Suppose that a function $\psi : \mathbb{C}^2 \times \Delta \to \mathbb{C}$ satisfies the condition: $$\psi(i\rho,\sigma;z) \notin \Xi$$ , for all real $\rho, \sigma \leq -|b-i\rho|^2$ /(2Reb) and all $z \in \Delta$ . If the function p(z) defined by $p(z) = b + b_1 z + b_2 z^2 + \cdots$ is analytic in $\Delta$ and if $$\psi(p(z), zp'(z); z) \in \Xi,$$ then $\operatorname{Re}(p(z)) > 0$ in $\Delta$ .

Lemma 1.2. [9] Let the function g(z) given by $$q(z) = 1 + c_1 z + c_2 z^2 + \cdots$$ be in the class $\mathcal{P}$ . Then, for any complex number $\mu$ $$|c_2 - \mu c_1^2| \le 2 \max\{1, |2\mu - 1|\}.$$ The result is sharp. The structure of the paper is the following. In Section 2, at first, we obtain a lower bound for the $\operatorname{Re} \frac{f(z)}{z}$ and by using it we get $\mathcal{S}^* \subset \mathcal{Q}(1/2)$ . In the sequel, we obtain the order of strongly starlikeness for the functions which belong to the class $\mathcal{SK}(\alpha)$ . In Section 3, sharp coefficient logarithmic inequality and sharp Fekete-Szegö inequality are obtained.

Theorem 2.1. Let $f \in A$ be in the class $SK(\alpha)$ and $0 \le \alpha \le 1$ . Then (2.1) $$\operatorname{Re}\left(\frac{f(z)}{z}\right) > \gamma(\alpha) := \frac{2\alpha^2 + 3\alpha + 9}{3(\alpha^2 + 3\alpha + 6)} \qquad (z \in \Delta).$$ That is means that $SK(\alpha) \subset Q(\gamma(\alpha))$ .

Corollary 2.1. Let $f \in A$ be defined by (1.1). Then $S^* \subset \mathcal{Q}(1/2)$ , i.e. $$\operatorname{Re}\left(\frac{zf'(z)}{f(z)}\right) > 0 \Rightarrow \operatorname{Re}\left(\frac{f(z)}{z}\right) > \frac{1}{2} \qquad (z \in \Delta).$$ We remark that in [4, 6], the authors with a different method have shown that $S^* \subset \mathcal{Q}(1/2)$ . By putting $z = re^{i\varphi}(r < 1), \varphi \in [0, 2\pi)$ and with a simple calculation, we have $$\begin{split} \widetilde{q}_{\alpha}(re^{i\varphi}) &= \frac{3}{3 + (\alpha - 3)re^{i\varphi} - \alpha r^2 e^{2i\varphi}} \\ &= \frac{9 + 3(\alpha - 3)r\cos\varphi - 3\alpha r^2\cos2\varphi - 3i[(\alpha - 3)r\sin\varphi - \alpha r^2\sin2\varphi]}{9 + (\alpha - 3)^2r^2 + \alpha^2r^4 + 2r(\alpha - 3)(3 - \alpha r^2)\cos\varphi - 6\alpha r^2\cos2\varphi}. \end{split}$$ Hence $$\left| \frac{\operatorname{Im}\{\widetilde{q}_{\alpha}(re^{i\varphi})\}}{\operatorname{Re}\{\widetilde{q}_{\alpha}(re^{i\varphi})\}} \right| = \left| \frac{3[(\alpha - 3)r\sin\varphi - \alpha r^{2}\sin2\varphi]}{9 + 3(\alpha - 3)r\cos\varphi - 3\alpha r^{2}\cos2\varphi} \right| (2.5) $$< \frac{3(3 - \alpha)r + |\alpha|r^{2}}{9 - 3(3 - \alpha)r - 3|\alpha|r^{2}} =: \phi(r) \qquad (r < 1, \quad -3 < \alpha \le 1).$$$$ For such r the curve $\widetilde{q}_{\alpha}(re^{i\varphi})$ , $\varphi \in [0, 2\pi)$ , has no loops and $\widetilde{q}_{\alpha}(re^{i\varphi})$ is univalent in $\Delta_r = \{z : |z| < r\}$ . Therefore (2.6) $$\left[ \frac{zf'(z)}{f(z)} \prec \widetilde{q}_{\alpha}(z), \quad z \in \Delta_r \right] \Leftrightarrow \left[ \frac{zf'(z)}{f(z)} \in \widetilde{q}_{\alpha}(\Delta_r), \quad z \in \Delta_r \right].$$ The above relations give the following theorem.

Theorem 2.2. Let $-1 < \alpha \le 1$ . If $f \in \mathcal{SK}(\alpha)$ , then f is strongly starlike of order $$\frac{2}{\pi}\arctan\left\{\frac{3(3-\alpha)+|\alpha|}{9-3(3-\alpha)-3|\alpha|}\right\},\,$$ in the unit disc $\Delta$ .

Theorem 3.1. Let $f \in \mathcal{A}$ and $-3 < \alpha \leq 1$ . If $f \in \mathcal{SK}(\alpha)$ , then there exists a function $w(z) \in \mathfrak{B}$ such that (3.2) $$\log \frac{f(z)}{z} = \int_0^z \frac{\widetilde{q}_{\alpha}(w(t)) - 1}{t} dt \qquad (z \in \Delta).$$

Corollary 3.1. Let $f \in A$ and $-3 < \alpha < 1$ . If $f \in SK(\alpha)$ , then (3.3) $$\log \frac{f(z)}{z} \prec \int_0^z \frac{\widetilde{q}_{\alpha}(t) - 1}{t} dt \qquad (z \in \Delta).$$ The celebrated de Branges' inequalities (the former Milin conjecture) for univalent functions f state that $$\sum_{n=1}^{k} (k-n+1)|\gamma_n|^2 \le \sum_{n=1}^{k} \frac{k+1-n}{n} \qquad (k=1,2,\ldots),$$ with equality if and only if $f(z) = e^{-i\theta}k(e^{i\theta}z)$ (see [1]). De Branges [1] used this inequality to prove the celebrated Bieberbach conjecture. Moreover, the de Branges' inequalities have also been the source of many other interesting inequalities involving logarithmic coefficients of $f \in \mathcal{S}$ such as (see [3]) $$\sum_{n=1}^{\infty} |\gamma_n|^2 \le \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.$$ Now, we have the following

Theorem 3.2. Let $f \in \mathcal{A}$ belongs to the class $\mathcal{SK}(\alpha)$ and $-3 < \alpha \leq 1$ . Then the logarithmic coefficients of f satisfy in the inequality (3.4) $$\sum_{n=1}^{\infty} |\gamma_n|^2 \le \frac{1}{4(3+\alpha)^2} \left[ \frac{3\pi^2}{2} + 6\alpha Li_2(-\alpha/3) + \alpha^2 Li_2(\alpha^2/9) \right],$$ where Li₂ is defined as following (3.5) $$Li_2(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^2} = \int_z^0 \frac{\ln(1-t)}{t} dt.$$ The inequality is sharp.

Theorem 3.3. Let $f \in A$ be a member of $SK(\alpha)$ . Then the logarithmic coefficients of f satisfy $$|\gamma_n| \le \frac{3-\alpha}{6n}$$ $(-3 < \alpha \le 1, n \ge 1).$

Theorem 3.4. Let that $f \in \mathcal{SK}(\alpha)$ , $-3 < \alpha \le 1$ and F is the kth root transform of f defined by (3.9). Then, for any complex number $\mu$ , $$(3.10) \qquad \left| b_{2k+1} - \mu b_{k+1}^2 \right| \le \frac{3-\alpha}{6k} \max \left\{ 1, \left| \frac{2\mu - 1}{k} \left( 1 - \frac{\alpha}{3} \right) - \frac{\alpha^2 - 3\alpha + 9}{6(3-\alpha)} \right| \right\}.$$ The result is sharp.

Corollary 3.2. (Fekete-Szegö inequality) Suppose that $f \in \mathcal{SK}(\alpha)$ . Then, for any complex number $\mu$ , $$(3.20) \left| a_3 - \mu a_2^2 \right| \le \frac{3 - \alpha}{6} \max \left\{ 1, \left| (2\mu - 1)(1 - \alpha/3) - \frac{\alpha^2 - 3\alpha + 9}{6(3 - \alpha)} \right| \right\}.$$ The result is sharp. If we take k = 1 and $\alpha = -1$ in Theorem 3.4, we get:

Corollary 3.3. Let f given by the form (1.1) be starlike function. Then $$(3.21) |a_3 - \mu a_2^2| \le \max\{2/3, |8(2\mu - 1)/9 - 7/12|\}.$$ The result is sharp. Taking k = 1 and $\alpha = 0$ in Theorem 3.4, we have:

Corollary 3.4. Let f given by the form (1.1) be in the class $S^*(1/2)$ . Then $$\left| a_3 - \mu a_2^2 \right| \le \frac{1}{2} \max \left\{ 1, \left| 2\mu - 3/2 \right| \right\}.$$ The result is sharp. It is well known that every function $f \in \mathcal{S}$ has an inverse $f^{-1}$ , defined by $f^{-1}(f(z)) = z, z \in \Delta$ and $$f(f^{-1}(w)) = w$$ $(|w| < r_0; r_0 > 1/4),$ where $$(3.23) f^{-1}(w) = w - a_2 w^2 + (2a_2^2 - a_3)w^3 - (5a_2^3 - 5a_2a_3 + a_4)w^4 + \cdots$$

Corollary 3.5. Let the function f, given by (1.1), be in the class $\mathcal{SK}(\alpha)$ . Also let the function $f^{-1}(w) = w + \sum_{n=2}^{\infty} b_n w^n$ be inverse of f. Then $$(3.24) |b_3| \le \frac{3-\alpha}{6} \max\left\{1, \left| \frac{5\alpha^2 - 33\alpha + 45}{6(3-\alpha)} \right| \right\}.$$

Definition 1.1. The function $f \in \mathcal{A}$ belongs to the class $\mathcal{SK}(\alpha)$ , $\alpha \in (-3,1]$ , if it satisfies the condition (1.7) $$\frac{zf'(z)}{f(z)} \prec \widetilde{q}_{\alpha}(z) \qquad (z \in \Delta),$$ where $\widetilde{q}_{\alpha}$ is given by (1.4). Since Re $\{\widetilde{q}_{\alpha}(z)\}$ > 9(1 + $\alpha$ )/2(3 + $\alpha$ )², therefore if $f \in \mathcal{SK}(\alpha)$ , then (1.8) $$\operatorname{Re}\left(\frac{zf'(z)}{f(z)}\right) > \frac{9(1+\alpha)}{2(3+\alpha)^2} \qquad (z \in \Delta).$$ This means that if $f \in \mathcal{SK}(\alpha)$ , then it is starlike of order $\gamma$ where $\gamma = 9(1+\alpha)/2(3+\alpha)^2$ . Also, $\mathcal{SK}(\alpha) \subset \mathcal{S}$ when $-1 \leq \alpha < 1$ , $\mathcal{SK}(0) \equiv \mathcal{S}^(1/2)$ , $\mathcal{SK}(1) \equiv \mathcal{S}^(9/16)$ and $\mathcal{SK}(-1) \equiv \mathcal{S}$ . We denote by $\mathcal{P}$ the well-known class of analytic functions p(z) with p(0) = 1 and Re(p(z)) > 0, $z \in \Delta$ . For the proof of our results, we need the following Lemmas.