Moduli difference of initial inverse logarithmic coefficients for starlike and convex functions

Theorem 2.1. Let $f \in \mathcal{S}_S^*$ be given by (1.1), then the following sharp inequality holds: $$|\Gamma_2| - |\Gamma_1| \le \frac{1}{2}.$$ The inequality is sharp. To prove our results, we need the following Lemma.

Lemma 2.1. [24] 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) $$\Psi_{+}(c_{1}, c_{2}) \leq \begin{cases} |4B_{2} + 2B_{3}| - 2B_{1}, & \text{if } |2B_{2} + B_{3}| \geq |B_{3}| + B_{1}, \\ 2|B_{3}|, & \text{otherwise.} \end{cases}$$ (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. Proof of Theorem 2.1. If $f \in \mathcal{S}_S^*$ be given by (1.1), then by the principle of subordination, there exists a Schwarz function $p(z) = 1 + \sum_{n=1}^{\infty} c_n z^n$ such that (2.2) $$\frac{2zf'(z)}{f(z) - f(-z)} = p(z).$$ By comparing coefficients of powers of z in both sides, we obtain $$(2.3) a_2 = \frac{1}{2}c_1, a_3 = \frac{1}{2}c_2.$$ Using (1.5) and (2.3), we obtain (2.4) $$|\Gamma_2| - |\Gamma_1| = \frac{1}{16} (|B_3 c_2 + B_2 c_1^2| - |B_1 c_1|) = \frac{1}{48} \Psi_+(c_1, c_2)$$ where $B_1 = 4$ , $B_2 = 3$ and $B_3 = -4$ . It is easy to see that the condition $|2B_2+B_3|=2<|B_3|+B_1=8$ holds. Therefore, by Lemma 2.1, we have $\Psi_+(c_1,c_2)\leq 2|B_3|$ . Then from (2.4) we have $$|\Gamma_2| - |\Gamma_1| \le \frac{2|-4|}{16} = \frac{1}{2}.$$ To show the equality, we consider the function $$f_1(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \cdots, \quad z \in \mathbb{D}.$$ Moreover, a simple computation yields $$\frac{2zf_1'(z)}{f_1(z) - f_1(-z)} = \frac{1+z}{1-z} =: p_1(z).$$ The function $$p_1(z) = \frac{1+z}{1-z} = 1 + 2z + 2z^2 + 2z^3 + \cdots$$ belongs to the Carathéodory class $\mathcal{P}$ . Hence, $f_1 \in \mathcal{S}_S^*$ . Now a simple computation shows that $$|\Gamma_2| - |\Gamma_1| = \frac{1}{2}.$$ This shows that the inequality in Theorem 2.1 is sharp. .  FIGURE 2. The graph exhibits the image domain of $f_1(\mathbb{D})$ . Next, we obtain the upper bound for $|\Gamma_2| - |\Gamma_1|$ when f belongs to the class $\mathcal{K}_S$ .

Theorem 2.2. If $f \in \mathcal{K}_S$ be given by (1.1), then the following sharp inequality holds: $$|\Gamma_2| - |\Gamma_1| \le \frac{1}{6}.$$ The inequality is sharp. Proof of Theorem 2.2. Let $f \in \mathcal{K}_S$ be given by (1.1), then by the principle of subordination, there exists a Schwarz function $p(z) = \sum_{n=1}^{\infty} c_n z^n$ such that $$\frac{2(zf'(z))'}{(f(z)-f(-z))'} = p(z).$$ By comparing coefficients of powers of z in both sides, we obtain $$(2.6) a_2 = \frac{1}{4}c_1, a_3 = \frac{1}{6}c_2.$$ Now using (1.5) together with (2.3), we have (2.7) $$|\Gamma_2| - |\Gamma_1| = \frac{1}{192} (|B_3 c_2 + B_2 c_1^2| - |B_1 c_1|) = \frac{1}{16} \Psi_+(c_1, c_2)$$ where $B_1 = 24$ , $B_2 = 9$ , and $B_3 = -16$ . For the upper bound, we observe that the condition $|2B_2 + B_3| = 2 < |B_3| + B_1 = 40$ is satisfied. Therefore, by Lemma (2.1), we have $\Psi_+(c_1, c_2) \le 2|B_3|$ . Then from (2.7) we have $$|\Gamma_2| - |\Gamma_1| \le \frac{2|-16|}{192} = \frac{1}{6}.$$ To show the equality, we consider the function $$f_2(z) = \frac{1}{2} \log \left( \frac{1+z}{1-z} \right) z + \frac{z^3}{3} + \frac{z^5}{5} + \cdots, \ z \in \mathbb{D}.$$ Moreover, a simple computation yields $$\frac{2(zf_2'(z))'}{(f_2(z)-f_2(-z))'} = \frac{1+z^2}{1-z^2} =: p_2(z).$$ The function $$p_2(z) = \frac{1+z^2}{1-z^2} = 1 + 2z^2 + 2z^4 + \cdots$$ belongs to the Carathéodory class $\mathcal{P}$ . Hence, $f_2 \in \mathcal{K}_S$ . A simple computation shows that $$|\Gamma_2| - |\Gamma_1| = \left| -\frac{1}{6} \right| - 0 = \frac{1}{6}.$$ This shows that the inequality in Theorem 2.2 is sharp. This completes the proof. We now establish the sharp lower and upper bounds of $|\Gamma_2| - |\Gamma_1|$ for the class $\mathcal{S}^*_{\mathcal{C}}$ .  FIGURE 3. The graph exhibits the image domain of $f_2(\mathbb{D})$ .

Theorem 2.3. If $f \in \mathcal{S}_{\mathbb{Q}}^*$ be given by (1.1), then the following sharp inequality holds: (2.8) $$-\frac{1}{\sqrt{10}} \le |\Gamma_2| - |\Gamma_1| \le \frac{1}{4}.$$ The inequalities are sharp. Proof of Theorem 2.3. Let $f \in \mathcal{S}_{\mathbb{Q}}^*$ . Then, in view of Definition 1.1, it follows that (2.9) $$\frac{zf'(z)}{f(z)} = w(z) + \sqrt{1 + w^2(z)},$$ where w is a Schwarz function with w(0) = 0 and $|w(z)| \le 1$ in $\mathbb{D}$ . Let $p \in \mathcal{P}$ . Then we can write (2.10) $$w(z) = \frac{p(z) - 1}{p(z) + 1}.$$ From (2.9) and (2.10), a simple computation shows that (2.11) $$a_2 = \frac{1}{2} c_1 \text{ and } a_3 = \frac{1}{16} c_1^2 + \frac{1}{4} c_2.$$ Using (1.5) together with (2.11), we see that $$(2.12) |\Gamma_2| - |\Gamma_1| = \frac{1}{32} \left( |B_3 c_2 + B_2 c_1^2| - |B_1 c_1| \right) := \frac{1}{32} \Psi_+(c_1, c_2)$$ where $B_1 = 8$ , $B_2 = 5$ , and $B_3 = -4$ . For the upper bound, we see that the condition $|2B_2 + B_3| = 6 < |B_3| + B_1 = 12$ . Therefore, by Lemma 2.1, we have $\Psi_+(c_1, c_2) \le 2|B_3|$ . Then from (2.12) we have $$|\Gamma_2| - |\Gamma_1| \le \frac{2|B_3|}{32} = \frac{1}{4}.$$ To show the equality, we consider the function $$f_3(z) = z \exp\left(\int_0^z \frac{t^2 + \sqrt{1 + t^4} - 1}{t} dt\right)$$ $$= \frac{\sqrt{2}z \exp\left(\frac{z^2 - 1 + \sqrt{1 + z^4}}{2}\right)}{\left(\sqrt{1 + z^4} + 1\right)^{\frac{1}{2}}}$$ $$= z + \frac{1}{2}z^3 + \dots, \quad z \in \mathbb{D}.$$ Moreover, a simple computation yields $$\frac{zf_3'(z)}{f_3(z)} = z^2 + \sqrt{1 + z^4} =: p_3(z).$$ The function $p_3(z)=z^2+\sqrt{1+z^4}$ is subordinate to a Carathéodory function. Indeed, letting $$p(z) = \frac{1+z^2}{1-z^2} = 1 + 2z^2 + 2z^4 + \dots \in \mathcal{P},$$ and using the relation w(z)=(p(z)-1)/(p(z)+1), we obtain $w(z)=z^2.$ Hence, $f_3\in\mathcal{S}_{\mathbb{Q}}^*$ . Then a simple computation shows that $$|\Gamma_2| - |\Gamma_1| = \frac{1}{4}.$$ This shows that the right-hand side inequality in Theorem 2.3 is sharp.  FIGURE 4. The graphs exhibit the image domains of $f_3(\mathbb{D})$ and $f_4(\mathbb{D})$ . We now consider the lower bound. Then (2.13) $$|\Gamma_2| - |\Gamma_1| = \frac{1}{32} \Psi_-(c_1, c_2),$$ where $\Psi_+(c_1, c_2) = -\Psi_-(c_1, c_2)$ . Since $B_4 = |4B_2 + 2B_3| = 12$ , it is easy to see that the inequality $B_1 \ge B_4 + 2|B_3|$ does not hold and the inequality $$B_1^2 = 64 < 2|B_3|(B_4 + 2|B_3|) = 160$$ holds. Hence, by Lemma 2.1, we obtain (2.14) $$\Psi_{-}(c_1, c_2) \le 2B_1 \sqrt{\frac{2|B_3|}{B_4 + 2|B_3|}} = 16\sqrt{\frac{2}{5}}.$$ Therefore from (2.13) and (2.14) we obtain the required inquality $$|\Gamma_2| - |\Gamma_1| \ge -\frac{1}{\sqrt{10}}.$$ Equality in the left-hand side of Theorem 2.3 is attained for the function $$p(z) = \frac{1 + 2Az + z^2}{1 - z^2}, \qquad A = \frac{2}{\sqrt{10}}.$$ The corresponding extremal function is given by $$f_4(z) = z \exp\left(\int_0^z \frac{w(t) + \sqrt{1 + w^2(t)} - 1}{t} dt\right),$$ where $w(z) = Az + z^2/1 - z^2$ . This function belongs to the class $\mathcal{S}_{\mathbb{Q}}^*$ and satisfies $|\Gamma_2| - |\Gamma_1| = -1/\sqrt{10}$ . Therefore, the lower bound is sharp. We now establish the sharp lower and upper estimates of $|\Gamma_2 - \Gamma_1|$ for the class $\mathcal{C}_{\mathbb{C}}$ .

Theorem 2.4. If $f \in \mathcal{C}_{\mathbb{Q}}$ be given by (1.1), then the following sharp inequality holds: $$(2.15) -\frac{4}{21} \le |\Gamma_2| - |\Gamma_1| \le \frac{1}{12}.$$ The inequalities are sharp. Proof of Theorem 2.4. Let $f \in \mathcal{C}_{\mathbb{Q}}$ . Then, in view of Definition 1.1, it follows that (2.16) $$1 + \frac{zf''(z)}{f'(z)} = w(z) + \sqrt{1 + w^2(z)}.$$ where w is a Schwarz function with w(0) = 0 and $|w(z)| \le 1$ in $\mathbb{D}$ . Let $p \in \mathcal{P}$ . Then we can write (2.17) $$w(z) = \frac{p(z) - 1}{p(z) + 1}.$$ From (2.16) and (2.17), a simple computation shows that (2.18) $$a_2 = \frac{1}{4}c_1 \text{ and } a_3 = \frac{1}{48}c_1^2 + \frac{1}{12}c_2.$$ Using (1.5) together with (2.18), we have (2.19) $$|\Gamma_2| - |\Gamma_1| = \frac{1}{192} (|B_3 c_2 + B_2 c_1^2| - |B_1 c_1|) = \frac{1}{16} \Psi_+(c_1, c_2)$$ where $B_1 = 24$ , $B_2 = 7$ , and $B_3 = -8$ . For the upper bound, the condition $|2B_2 + B_3| = 6 < |B_3| + B_1 = 32$ holds; thus, by Lemma 2.1, we obtain $$|\Gamma_2| - |\Gamma_1| \le \frac{2|-8|}{192} = \frac{1}{12}.$$ To show the equality, we consider the function $$f_5(z) = \int_0^z \exp\left(\int_0^\zeta \frac{t^2 + \sqrt{1 + t^4} - 1}{t} dt\right) d\zeta$$ $$= \int_0^z \frac{\sqrt{z} \exp\left(\frac{\zeta^2 - 1 + \sqrt{1 + \zeta^4}}{2}\right)}{\left(\sqrt{1 + \zeta^4} + 1\right)^{\frac{1}{2}}}$$ $$= z + \frac{1}{6}z^3 + \frac{1}{20}z^5 + \cdots, \quad z \in \mathbb{D}.$$ Moreover, a simple computation yields $$1 + \frac{zf_5''(z)}{f_5'(z)} = z^2 + \sqrt{1 + z^4} =: p_4(z).$$ The function $p_4(z)=z^2+\sqrt{1+z^4}$ is subordinate to a Carathéodory function. Indeed, letting $$p(z) = \frac{1+z^2}{1-z^2} = 1 + 2z^2 + 2z^4 + \dots \in \mathcal{P},$$ and using the relation $$w(z) = \frac{p(z) - 1}{p(z) + 1},$$ we obtain $w(z) = z^2$ . Hence, $$1 + \frac{zf_5''(z)}{f_5'(z)} = w(z) + \sqrt{1 + w^2(z)},$$ and therefore $f_5 \in \mathcal{C}_{\mathbb{Q}}$ . A simple computation shows that $$|\Gamma_2| - |\Gamma_1| = \frac{1}{12}.$$ We now consider the lower bound. Then (2.20) $$|\Gamma_2| - |\Gamma_1| = \frac{1}{102} \Psi_-(c_1, c_2),$$ where $\Psi_+(c_1,c_2) = -\Psi_-(c_1,c_2)$ . Since $B_4 = |4B_2 + 2B_3| = 12$ , the inequality $B_1 \geq B_4 + 2|B_3|$ is not satisfied. Moreover, $2|B_3|(B_4 + 2|B_3|) = 448$ , and hence the inequality $B_1^2 \leq 2|B_3|(B_4 + 2|B_3|)$ is also not satisfied. Hence, by Lemma 2.1, we obtain (2.21) $$\Psi_{-}(c_1, c_2) \le \left(2|B_3| + \frac{B_1^2}{B_4 + 2|B_3|}\right) = \frac{256}{7}.$$  FIGURE 5. The graphs exhibit the image domains of $f_5(\mathbb{D})$ and $f_6(\mathbb{D})$ . Therefore from (2.20) and (2.21) we obtain the required inquality $$|\Gamma_2| - |\Gamma_1| \ge -\frac{4}{21}.$$ The equality in the left–hand side of Theorem 2.3 for the class $\mathcal{C}_{\mathbb{C}}$ is attained for $$p(z) = \frac{1 + 2Az + z^2}{1 - z^2}, \qquad A = \frac{4}{7}.$$ The corresponding extremal function $$f_6(z) = \int_0^z \exp\left(\int_0^\zeta \frac{w(t) + \sqrt{1 + w(t)^2} - 1}{t} dt\right) d\zeta,$$ where $$w(z) = \frac{Az + z^2}{1 + Az},$$ belongs to $\mathcal{C}_{\mathbb{Q}}$ and satisfies $|\gamma_2| - |\gamma_1| = -4/21$ . Hence, the lower bound is sharp. This completes the proof. Acknowledgment: The authors would like thank the anonymous referee for his/her elaborate comments and valuable suggestions which improve significantly the presentation of the paper. The first author is supported by SERB File No. SUR/2022/002244, Govt. of India, The second author is supported by UGC-JRF (NTA Ref. No.: 201610135853), New Delhi, India.

Definition 1.1. (see [6, p. 43, Theorem 2.12]). Let f and g be two analytic functions in $\mathbb{D}$ . Then f is said to be subordinate to g, written as $f \prec g$ or $f(z) \prec g(z)$ , if there exists a function $\omega$ , analytic in $\mathbb{D}$ with $\omega(0) = 0$ , $|\omega(z)| < 1$ , and $$f(z) = g(\omega(z))$$ for $z \in \mathbb{D}$ . Moreover, if g is univalent in $\mathbb{D}$ and f(0) = g(0), then $f(\mathbb{D}) \subset g(\mathbb{D})$ . In [18], Raina and Sokol introduced the class $\mathcal{S}^*_{\mathcal{C}}$ , defined by $$\mathcal{S}_{\mathbb{C}}^* := \left\{ f \in \mathcal{S} : \left| \left( \frac{zf'(z)}{f(z)} \right)^2 - 1 \right| \le 2 \left| \frac{zf'(z)}{f(z)} \right|, \quad z \in \mathbb{D} \right\}.$$ Geometrically, a function $f \in \mathcal{S}_{\mathbb{Q}}^*$ is such that, for any $z \in \mathbb{D}$ , the quantity $\frac{zf'(z)}{f(z)}$ lies in the region bounded by a lune. This region is given by $$\Omega = \left\{ w \in \mathbb{C} : |w^2 - 1| \le 2|w| \right\}.$$ By using the definition of subordination, the class $\mathcal{S}^*_{\mathbb{Q}}$ can be equivalently defined as $$\mathcal{S}_{\mathbb{Q}}^* \ = \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec z + \sqrt{1+z^2} = q(z), \quad z \in \mathbb{D} \right\},$$ where the branch of the square root is chosen so that q(0) = 1. Similarly, the class $\mathcal{C}^*_{\mathbb{Q}}$ of convex functions associated with the lune is defined by $$\mathcal{C}_{\mathbb{C}} := \left\{ f \in \mathcal{S} : 1 + \frac{zf''(z)}{f'(z)} \prec q(z), \quad z \in \mathbb{D} \right\}.$$ The class $\mathcal{S}^_{\mathbb{Q}}$ has been extensively investigated by several authors [18, 19, 20]. The coefficient bounds and the sharp Fekete–Szegö inequality for this class were established by Raina and Sokol (see [19, 20]). Sharma et al. [23] examined certain differential subordinations related to $\mathcal{S}^_{\mathbb{Q}}$ . In [21], Riaz and Raza established results concerning the third Hankel determinant for starlike and convex functions associated with lune. Integral representations and sufficient conditions for functions in $\mathcal{S}^_{\mathbb{Q}}$ were provided by Raina et al.* (see [18]).  FIGURE 1. The figure focuses strictly on the Lune domain $\mathcal{L}$ defined by the inequality $|w^2-1| \leq 2|w|$ . The region is bounded by two symmetric arcs that meet at the vertical vertices i and -i. On the real axis, the right-hand lobe extends from $1-\sqrt{2}\approx -0.414$ to $1+\sqrt{2}\approx 2.414$ . Due to symmetry, the left-hand lobe spans from $-(1+\sqrt{2})$ to $-(1-\sqrt{2})$ . This region is the range of the function $q(z)=z+\sqrt{1+z^2}$ for z in the unit disk $\mathbb{D}$ . In geometric function theory, a function f belongs to the class $\mathcal{S}_{\mathbb{Q}}^*$ if the quantity $\frac{zf'(z)}{f(z)}$ takes values within this domain. In 1985, de Branges [5] solved the famous Bieberbach conjecture, by showing that if $f \in \mathcal{S}$ of the form (1.2), then $|a_n| \leq n$ for $n \geq 2$ with equality holds for the Koebe function $k(z) := z/(1-z)^2$ or its rotations. It was therefore natural to ask if for $f \in \mathcal{S}$ , the inequality $||a_{n+1}| - |a_n|| \leq 1$ is true when $n \geq 2$ . This problem was first studied by Goluzin [7] with an aim to solve the Bieberbach conjecture. In 1963, Hayman [9] proved that $||a_{n+1}| - |a_n|| \leq A$ for $f \in \mathcal{S}$ , where $A \geq 1$ is an absolute constant and the best known estimate as of now is 3.61 due to Grinspun [8]. On the other hand, for the class $\mathcal{S}$ , the sharp bound is known only for n = 2 (see [6], Theorem 3.11), namely $$-1 \le |a_3| - |a_2| \le 1.029\dots$$ Similarly, for functions $f \in \mathcal{S}^*$ , Pommerenke [16] has conjectured that $||a_{n+1}| - |a_n|| \le 1$ which was proved later in 1978 by Leung [10]. For convex functions, Li and Sugawa [11] investigated the sharp upper bound of $|a_{n+1}| - |a_n|$ for $n \ge 2$ , and sharp lower bounds for n = 2, 3. Let $f \in \mathcal{S}$ and let $F = f^{-1}$ be its inverse, which has the series representation $$F(w) = w + \sum_{n=2}^{\infty} A_n w^n,$$ valid in a neighborhood of the origin. Using the identity $f(f^{-1}(w)) = w$ and comparing coefficients, we obtain $$(1.3) A_2 = -a_2 \text{ and } A_3 = 2a_2^2 - a_3.$$ Inverse functions of univalent mappings have been widely studied. The inverse functions are studied by several authors in different perspectives (see, for instance, [3, 24, 26] and references therein). Recently, Sim and Thomas [24, 25] obtained sharp upper and lower bounds on the difference of the moduli of successive inverse coefficients for subclasses of univalent functions. For other coefficient problems concerning logarithmic coefficients, readers are referred to the articles [2, 1, 14, 12, 13] and references therein. In particular, sharp bounds for differences of successive inverse coefficients were recently established for several subclasses. The logarithmic coefficients $\gamma_n$ of f are defined by $$\log \frac{f(z)}{z} = 2 \sum_{n=1}^{\infty} \gamma_n z^n, \quad z \in \mathbb{D}.$$ The logarithmic coefficients $\gamma_n$ are fundamental in the theory of univalent functions, although only a few sharp bounds are known. Their relevance to the Bieberbach conjecture was highlighted by Milin [15], who conjectured that for $f \in \mathcal{S}$ and $n \geq 2$ , $$\sum_{m=1}^{n} \sum_{k=1}^{m} \left( k |\gamma_k|^2 - \frac{1}{k} \right) \le 0.$$ This conjecture was later proved by de Branges, leading to the solution of the Bieberbach conjecture. For the Koebe function $k(z) = z/(1-z)^2$ , one has $\gamma_n = 1/n$ , but the estimate $|\gamma_n| \leq 1/n$ does not hold in general for functions in $\mathcal{S}$ . Recent studies have therefore focused on logarithmic coefficients in $\mathcal{S}$ and its subclasses. Inverse logarithmic coefficients were introduced by Ponnusamy et al. [17]. They are defined by (1.4) $$\log \frac{f^{-1}(w)}{w} = 2\sum_{n=1}^{\infty} \Gamma_n w^n, \qquad |w| < \frac{1}{4}.$$ Differentiating (1.4) and using (1.3), we obtain (1.5) $$\Gamma_1 = -\frac{1}{2}a_2 \text{ and } \Gamma_2 = -\frac{1}{2}a_3 + \frac{3}{4}a_2^2.$$ Ponnusamy et al. [17] later established sharp bounds for the inverse logarithmic coefficients. In particular, for $f \in \mathcal{S}$ , $$|\Gamma_n| \le \frac{1}{2n} \binom{2n}{n}, \quad n \in \mathbb{N}.$$ Recently, Allu and Shaji (see [4]) obtained sharp lower and upper bounds for the quantity $|\Gamma_2| - |\Gamma_1|$ for functions f belonging to the classes $\mathcal{S}$ , $\mathcal{S}$ , $\mathcal{C}$ , $\mathcal{S}^_{\alpha}$ , $\mathcal{C}_{\alpha}$ , $\mathcal{S}^(\alpha)$ , $\mathcal{C}(\alpha)$ , $\mathcal{G}(\nu)$ , $\mathcal{F}_0(\lambda)$ , $\mathcal{S}^_{\gamma}(\alpha)$ , and $\mathcal{C}_{\gamma}(\alpha)$ . In [22], the authors have established results for moduli difference of initial inverse coefficients of Bazilevič functions. The primary objective of this paper is to derive sharp lower and upper bounds for $|\Gamma_2| - |\Gamma_1|$ for functions f belonging to a given subclass of A.