Moduli difference of inverse logarithmic coefficients of univalent functions

Lemma 2.2. [30] Let $B_1$ , $B_2$ , and $B_3$ be numbers such that $B_1 > 0$ , $B_2 \in \mathbb{C}$ , and $B_3 \in \mathbb{R}$ . Let $p \in \mathcal{P}$ be of the form (2.1). Define $\Psi_+(c_1, c_2)$ and $\Psi_-(c_1, c_2)$ by $$\Psi_{+}(c_1, c_2) = |B_2 c_1^2 + B_3 c_2| - |B_1 c_1|,$$ and $$\Psi_{-}(c_1, c_2) = -\Psi_{+}(c_1, c_2).$$ Then (2.2) $$\Psi_{+}(c_{1}, c_{2}) \leq \begin{cases} |4B_{2} + 2B_{3}| - 2B_{1}, & if |2B_{2} + B_{3}| \geq |B_{3}| + B_{1}, \\ 2|B_{3}|, & otherwise, \end{cases}$$ and (2.3) $$\Psi_{-}(c_{1}, c_{2}) \leq \begin{cases} 2B_{1} - B_{4}, & \text{if } B_{1} \geq B_{4} + 2|B_{3}|, \\ 2B_{1}\sqrt{\frac{2|B_{3}|}{B_{4} + 2|B_{3}|}}, & \text{if } B_{1}^{2} \leq 2|B_{3}|(B_{4} + 2|B_{3}|), \\ 2|B_{3}| + \frac{B_{1}^{2}}{B_{4} + 2|B_{3}|}, & \text{otherwise,} \end{cases}$$ where $B_4 = |4B_2 + 2B_3|$ . All inequalities in (2.2) and (2.3) are sharp. Our main aim of this paper is to estimate the sharp Lower and upper bounds of $|\Gamma_2| - |\Gamma_1|$ for functions f belong to $\mathcal{S}$ , $\mathcal{S}^*$ , $\mathcal{C}$ , $\mathcal{S}^*_{\alpha}$ , $\mathcal{C}_{\alpha}$ , $\mathcal{S}^*(\alpha)$ , $\mathcal{C}(\alpha)$ , $\mathcal{G}(\nu)$ , $\mathcal{F}_0(\lambda)$ , $\mathcal{S}^*_{\gamma}(\alpha)$ and $\mathcal{C}_{\gamma}(\alpha)$ .

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Lemma 2.2. [30] Let  $B_1$ ,  $B_2$ , and  $B_3$  be numbers such that  $B_1 > 0$ ,  $B_2 \in \mathbb{C}$ , and  $B_3 \in \mathbb{R}$ . Let  $p \in \mathcal{P}$  be of the form (2.1). Define  $\Psi_+(c_1, c_2)$  and  $\Psi_-(c_1, c_2)$  by

$$\Psi_{+}(c_1, c_2) = |B_2 c_1^2 + B_3 c_2| - |B_1 c_1|,$$

and

$$\Psi_{-}(c_1, c_2) = -\Psi_{+}(c_1, c_2).$$

Then

(2.2) 
$$\Psi_{+}(c_{1}, c_{2}) \leq \begin{cases} |4B_{2} + 2B_{3}| - 2B_{1}, & if |2B_{2} + B_{3}| \geq |B_{3}| + B_{1}, \\ 2|B_{3}|, & otherwise, \end{cases}$$

and

(2.3) 
$$\Psi_{-}(c_{1}, c_{2}) \leq \begin{cases} 2B_{1} - B_{4}, & \text{if } B_{1} \geq B_{4} + 2|B_{3}|, \\ 2B_{1}\sqrt{\frac{2|B_{3}|}{B_{4} + 2|B_{3}|}}, & \text{if } B_{1}^{2} \leq 2|B_{3}|(B_{4} + 2|B_{3}|), \\ 2|B_{3}| + \frac{B_{1}^{2}}{B_{4} + 2|B_{3}|}, & \text{otherwise,} \end{cases}$$

where  $B_4 = |4B_2 + 2B_3|$ . All inequalities in (2.2) and (2.3) are sharp.

Our main aim of this paper is to estimate the sharp Lower and upper bounds of  $|\Gamma_2| - |\Gamma_1|$ for functions f belong to  $\mathcal{S}$ ,  $\mathcal{S}^$ ,  $\mathcal{C}$ ,  $\mathcal{S}^_{\alpha}$ ,  $\mathcal{C}_{\alpha}$ ,  $\mathcal{S}^(\alpha)$ ,  $\mathcal{C}(\alpha)$ ,  $\mathcal{G}(\nu)$ ,  $\mathcal{F}_0(\lambda)$ ,  $\mathcal{S}^_{\gamma}(\alpha)$  and  $\mathcal{C}_{\gamma}(\alpha)$ .

Theorem 3.1. Let $f \in \mathcal{S}$ of the form (1.1), then $$|\Gamma_2| - |\Gamma_1| \le \frac{1}{2}$$ and $$|\Gamma_2| - |\Gamma_1| \ge \begin{cases} -\frac{1}{2} & \text{if } |a_2| \le 1, \\ -0.6353..., & \text{if } |a_2| > 1. \end{cases}$$

Theorem 3.2. Let $\alpha \in (0,1]$ . If $f \in \mathcal{S}_{\alpha}^*$ given by (1.1), then the following sharp inequalities holds. $$(3.4) -\frac{\alpha}{\sqrt{1+3\alpha}} \le |\Gamma_2| - |\Gamma_1| \le \frac{\alpha}{2}.$$

Corollary 3.3. For every $f \in \mathcal{S}^*$ of the form (1.1), we have (3.8) $$-\frac{1}{2} \le |\Gamma_2| - |\Gamma_1| \le \frac{1}{2}.$$ Both inequalities are sharp. Next, we obtain the sharp lower and upper bounds for $|\Gamma_2| - |\Gamma_1|$ when f belongs to the class $\mathcal{C}_{\alpha}$ .

Theorem 3.4. Let $\alpha \in (0,1]$ . If $f \in \mathcal{C}_{\alpha}$ given by (1.1), then the following sharp inequalities holds; $$(3.9) |\Gamma_2| - |\Gamma_1| \le \frac{\alpha}{12}$$ and (3.10) $$|\Gamma_{2}| - |\Gamma_{1}| \ge \begin{cases} -\frac{1}{4}\alpha(2 - \alpha) & \text{if } 0 < \alpha \le \frac{1}{3}, \\ -\frac{\alpha}{6}\left(\frac{6\alpha + 3}{6\alpha + 14}\right) & \text{if } \frac{1}{3} < \alpha < \frac{5}{6}, \\ -\frac{\alpha}{\sqrt{4 + 6\alpha}} & \text{if } \frac{5}{6} \le \alpha \le 1. \end{cases}$$

Corollary 3.5. For every $f \in \mathcal{C}$ of the form (1.1), we have (3.15) $$-\frac{1}{\sqrt{10}} \le |\Gamma_2| - |\Gamma_1| \le \frac{1}{6}.$$ Both inequalities are sharp.

Theorem 3.6. Let $f \in \mathcal{G}(\nu)$ of the form (1.1). Then the following sharp estimate holds. $$(3.16) |\Gamma_2| - |\Gamma_1| \le \frac{\nu}{12}$$ (3.17) $$|\Gamma_2| - |\Gamma_1| \ge \begin{cases} \frac{\nu}{12} \left( \frac{10\nu + 34}{5\nu + 8} \right) & \text{if } \frac{1}{5} \le \nu \le 1, \\ \frac{\nu}{\sqrt{5\nu + 8}}, & \text{if } \frac{1}{5} \le \nu \le 1. \end{cases}$$

Theorem 3.7. Let $1/2 \le \lambda \le 1$ . For every $f \in \mathcal{F}_0(\lambda)$ be of the form (1.1), then the following sharp inequalities holds. (3.21) $$-\frac{1+2\lambda}{2\sqrt{5+10\lambda}} \le |\Gamma_2| - |\Gamma_1| \le \frac{\nu}{12}.$$

Theorem 3.8. Let $-\pi/2 < \gamma < \pi/2$ and $0 \le \alpha < 1$ . For every $f \in \mathcal{S}_{\gamma}^*(\alpha)$ of the form (1.1), the following sharp inequalities holds. $$-\frac{(1-\alpha)\cos\gamma}{\sqrt{|\eta|+1}} \le |\Gamma_2| - |\Gamma_1| \le \frac{(1-\alpha)\cos\gamma}{2}$$ where $\eta = 4(1 - \alpha)\mu - 1$ with $\mu = e^{i\gamma}\cos\gamma$ .

Corollary 3.9. Let ⁰ ^≤ α < ¹ and ^f ∈ S^∗ (α) of the form [\(1.1\)](#page-0-0). Then $$(3.32) |\Gamma_2| - |\Gamma_1| \le \frac{1 - \alpha}{2}$$ and (3.33) $$|\Gamma_2| - |\Gamma_1| \ge \begin{cases} -\frac{\sqrt{1-\alpha}}{2} & \text{if } 0 \le \alpha \le \frac{3}{4}, \\ -\frac{1-\alpha}{\sqrt{4\alpha-2}} & \text{if } \frac{3}{4} \le \alpha < 1, \end{cases}$$ Next, we obtain the sharp lower and upper bounds for |Γ2| − |Γ1| when f belongs to the class Cγ(α).

Theorem 3.10. Let −π/2 < γ < π/2 and 0 ≤ α < 1. For every f ∈ Cγ(α) of the form [\(1.1\)](#page-0-0), the following sharp inequality holds. $$(3.34) |\Gamma_2| - |\Gamma_1| \le \frac{(1-\alpha)\cos\gamma}{6}$$ and $$(3.35) |\Gamma_2| - |\Gamma_1| \ge \begin{cases} \frac{(\alpha - 1)\cos\gamma}{12} (6 - |\beta|), & \text{if } |\beta| \le 1, \\ \left(\frac{(\alpha - 1)\cos\gamma}{12}\right) \left(2 + \frac{9}{|\beta| + 2}\right), & \text{if } 1 \le |\beta| \le 5/2, \\ \left(\frac{(\alpha - 1)\cos\gamma}{2}\right) \left(\sqrt{\frac{2}{|\beta| + 2}}\right), & \text{if } |\beta| \ge 5/2. \end{cases}$$ where β = 5(1 − α)µ − 2 with µ = e iγ cos γ. Proof. Let f ∈ Cγ(α), then there exists a function p ∈ P such that (3.36) $$p(z) = \frac{1}{1-\alpha} \left\{ \frac{1}{\cos \gamma} \left( e^{-i\gamma} \left( 1 + \frac{zf''(z)}{f'(z)} \right) + i \sin \gamma \right) - \alpha \right\}.$$ Equating the coefficients of z ⁿ on both the sides of [\(3.36\)](#page-12-0) for n = 1, 2, we obtain (3.37) $$a_2 = \frac{(1-\alpha)\mu}{2}c_1 \text{ and } a_3 = \frac{(1-\alpha)\mu}{6}\left((1-\alpha)\mu c_1^2 + c_2\right),$$ where $\mu = e^{i\gamma}\cos\gamma$ . Now using (1.10), we obtain (3.38) $$|\Gamma_2| - |\Gamma_1| = \frac{(1-\alpha)\cos\gamma}{12} \left( |c_2 - \frac{5}{4}(1-\alpha)\mu c_1^2| - |3c_1| \right)$$ $$= \frac{(1-\alpha)\cos\gamma}{12} \left( |B_3c_2 + B_2c_1^2| - |B_1c_1| \right),$$ where $$B_1 := 3, B_2 := -\frac{5}{4}(1 - \alpha)\mu$$ , and $B_3 = 1$ . For the upper bound, it is easy to see that the first condition $|2B_2 + B_3| \ge |B_3| + B_1$ is not satisfied since $|5(1-\alpha)\mu - 2| \le 3$ . By using Lemma 2.2 and the equation (3.38), we obtain that $$|\Gamma_2| - |\Gamma_1| \le \frac{(1-\alpha)\cos\gamma}{6}.$$ This proves the inequality (3.34). We next prove the lower bound in (3.35) by checking the condition of Lemma 2.2 for $\Psi_{-}(c_1, c_2)$ . Note that the inequality $B_1 \geq B_4 + 2|B_3|$ is true for $|5(1-\alpha)\mu - 2| \leq 1$ and the inequality $B_1^2 \leq 2|B_3|(B_4+2|B_3|)$ is true for $|5(1-\alpha)\mu - 2| \geq 5/2$ . Thus, by applying Lemma 2.2, we obtain $$\Psi_{-}(c_{1}, c_{2}) \leq \begin{cases} 6 - |\beta|, & \text{if } |\beta| \leq 1, \\ 2 + \frac{9}{|\beta| + 2}, & \text{if } 1 \leq |\beta| \leq 5/2, \\ 6\left(\sqrt{\frac{2}{|\beta| + 2}}\right), & \text{if } |\beta| \geq 5/2. \end{cases}$$ where $\beta = 5(1 - \alpha)\mu - 2$ . By Substituting the above inequality in (3.38) we obtain the required inequality (3.35). Equality holds in (3.34) when f is defined by (3.36) with $p(z) = (1 + z^2)/(1 - z^2)$ . For $|\beta| \ge 5/2$ , equality holds for the function f defined by (3.36) with (3.39) $$p(z) = \frac{1 + q_1(q_2 + 1)z + q_2z^2}{1 + q_1(q_2 - 1) - q_2z^2},$$ where $q_1 = 1/\sqrt{1+|\beta|}$ and $q_2 = e^{i\arg\beta}$ . For $|\beta| \le 5/4$ , equality holds for the function f given by (3.36) where p(z) is of the form (3.39) with $q_1 = 3/2(1+|\beta|)$ and $q_2 = e^{i\arg\beta}$ . This completes the proof of this theorem. If we put $\gamma = 0$ in Theorem 3.10, then we obtain the following result for the class of convex function of order $\alpha$ .

Corollary 3.11. Let $0 \le \alpha < 1$ and $f \in C(\alpha)$ of the form (1.1). Then (3.40) $$|\Gamma_2| - |\Gamma_1| \le \frac{1}{6}(1 - \alpha)$$ and , , (3.41) $$|\Gamma_{1}| - |\Gamma_{2}| \leq \begin{cases} \sqrt{\left(\frac{1-\alpha}{10}\right)}, & \text{if } 0 \leq \alpha \leq \frac{1}{10}, \\ \frac{1}{60}(19-10\alpha), & \text{if } \frac{1}{10} \leq \alpha \leq \frac{2}{5}, \\ \frac{1}{12}(1-\alpha)(3+5\alpha), & \text{if } \frac{2}{5} \leq \alpha \leq \frac{3}{5}, \\ \frac{1}{12}(1-\alpha)(9-5\alpha), & \text{if } \frac{3}{5} \leq \alpha \leq \frac{4}{5}, \\ \frac{(1-\alpha)(10\alpha+7)}{12(5\alpha-1)}, & \text{if } \frac{4}{5} \leq \alpha \leq 1. \end{cases}$$ All the inequalities are sharp. Acknowledgement: The research of the first named author is supported by SERB-CRG, Govt. of India and the second named author's research work is supported by UGC-SRF.

Definition 1.1. A locally univalent function $f \in \mathcal{A}$ is said to belong to $\mathcal{G}(\nu)$ for some $\nu > 0$ , if it satisfies the condition $$\operatorname{Re}\left(1 + \frac{zf''(z)}{f'(z)}\right) < 1 + \frac{\nu}{2}, \quad z \in \mathbb{D}.$$ In 1941, Ozaki [21] introduced the class $\mathcal{G}(1) =: \mathcal{G}$ and proved that functions in $\mathcal{G}$ are univalent in $\mathbb{D}$ . Later, Umezawa [38] studied the class $\mathcal{G}$ and showed that this class contains the class of functions convex in one direction. Moreover, functions in $\mathcal{G}$ are proved to be starlike in $\mathbb{D}$ (see [25], [28]). Thus, the class $\mathcal{G}(\nu)$ is included in $\mathcal{S}^*$ whenever $\nu \in (0,1]$ . It can be easily seen that functions in $\mathcal{G}(\nu)$ are not necessarily univalent in $\mathbb{D}$ if $\nu > 1$ .

Definition 1.2. For $-1/2 < \lambda \le 1$ , the class $\mathcal{F}(\lambda)$ defined by $$\mathcal{F}(\lambda) = \left\{ f \in \mathcal{A} : \operatorname{Re}\left(1 + \frac{zf''(z)}{f'(z)}\right) > \frac{1}{2} - \lambda \text{ for } z \in \mathbb{D} \right\}.$$ We note that clearly $\mathcal{F}(1/2) =: \mathcal{C}$ is the usual class of convex functions. Moreover, for $\lambda = 1$ , we obtain the class $\mathcal{F}(1) =: \mathcal{C}(-1/2)$ which considered by many researcher in the recent years (see [?,4,19,26]). Also, functions in $\mathcal{C}(-1/2)$ are not necessarily starlike but are convex in some direction and so are close-to-convex. Here, we recall that a function $f \in \mathcal{A}$ is called close-to-convex if $f(\mathbb{D})$ is close-to-convex domain, i.e. the complement of $f(\mathbb{D})$ in $\mathcal{C}$ is the union of closed half lines with pairwise disjoint interiors. Pfaltzgraff et al. [23] has proved that $F(\lambda)$ contains non-starlike functions for all $1/2 < \lambda \leq 0$ . Next we will discuss about the family of spirallike functions.

Definition 1.3. The family $S_{\gamma}(\alpha)$ of $\gamma$ -spirallike functions of order $\alpha$ is defined by $$S_{\gamma}(\alpha) = \left\{ f \in \mathcal{A} : \operatorname{Re}\left(e^{-i\gamma} \frac{zf'(z)}{f(z)}\right) > \alpha \cos \gamma \text{ for } z \in \mathbb{D} \right\},$$ where $\alpha \in [0,1)$ and $\gamma \in (-\pi/2,\pi/2)$ . Each function in $S_{\gamma}(\alpha)$ is univalent in $\mathbb{D}$ (see [17]). Clearly, $S_{\gamma}(\alpha) \subset S_{\gamma}(0) \subset S$ whenever $0 \leq \alpha < 1$ . Functions in $S_{\gamma}(0)$ are called $\gamma$ -spirallike, but they do not necessarily belong to the starlike family $S$ . The class $S_{\gamma}(0)$ was introduced by Špaček [33]. Moreover, $S_0(\alpha) =: S^(\alpha)$ is Robertson's class of functions that are starlike functions of order $\alpha$ , and $S^(0) = S$ is the class of starlike functions. The class $S^*(\alpha)$ is meaningful even if $\alpha < 0$ , although univalency will be destroyed in this situation. We consider another family of functions that includes the class of convex functions as a proper subfamily.

Definition 1.4. The family $C_{\gamma}(\alpha)$ of $\gamma$ -convex functions of order $\alpha$ is defined by $$C_{\gamma}(\alpha) = \left\{ f \in \mathcal{A} : \operatorname{Re}\left(e^{-i\gamma}\left(1 + \frac{zf''(z)}{f'(z)}\right)\right) > \alpha \cos \gamma \right\}$$ where $0 \le \alpha < 1$ and $-\pi/2 < \gamma < \pi/2$ . We may set $C_0(\alpha) =: C(\alpha)$ which consists of the normalized convex functions of order $\alpha$ . Function in $C_{\gamma}(0) =: C_{\gamma}$ need not be univalent in $\mathbb{D}$ for general values of $\gamma$ ( $|\gamma| < \pi/2$ ). For example, the function $f(z) = i(1-z)^i - i$ is known to belong to $C_{\pi/4} \setminus S$ . Robertson [29] has shown that $f \in C_{\gamma}$ is univalent if $0 < \cos \gamma \le 0.2315 \cdots$ . Finally, Pfaltzgraff [22] has shown that $f \in C_{\gamma}$ is univalent whenever $0 < \cos \gamma \le 1/2$ . This settles the improvement of the range of $\gamma$ for which $f \in C_{\gamma}$ is univalent. On the other hand, in [32] it was also shown that functions in $C_{\gamma}$ which satisfy f''(0) = 0 are univalent for all real values of $\gamma$ with $|\gamma| < \pi/2$ . For the recent study of the class for particular values of $\alpha$ and $\gamma$ , we refer to [5].