On Coefficient problems for classes S_e and C_e

Lemma 2.1. [\[6,](#page-25-12) Lemma 2.4] If p ∈ P is of the form [\(2.1\)](#page-2-1), then $$c_1 = 2\tau_1, \tag{2.2}$$ $$c_2 = 2\tau_1^2 + 2(1 - \tau_1^2)\tau_2 \tag{2.3}$$ and $$c_3 = 2\tau_1^3 + 4(1 - \tau_1^2)\tau_1\tau_2 - 2(1 - \tau_1^2)\tau_1\tau_2^2 + 2(1 - \tau_1^2)(1 - |\tau_2|^2)\tau_3$$ (2.4) for some τ1, τ2, τ³ ∈ D := {z ∈ C : |z| ≤ 1}. For τ¹ ∈ T := {z ∈ C : |z| = 1}, there is a unique function p ∈ P with c¹ as in [\(2.2\)](#page-3-0), namely $$p(z) = \frac{1 + \tau_1 z}{1 - \tau_1 z}, \quad z \in \mathbb{D}.$$ For τ¹ ∈ D and τ² ∈ T, there is a unique function p ∈ P with c¹ and c² as in [\(2.2\)](#page-3-0) and [\(2.2\)](#page-3-1), namely $$p(z) = \frac{1 + (\overline{\tau}_1 \tau_2 + \tau_1)z + \tau_2 z^2}{1 + (\overline{\tau}_1 \tau_2 - \tau_1)z - \tau_2 z^2}, \quad z \in \mathbb{D}.$$ For τ1, τ² ∈ D and τ³ ∈ T, there is a unique function p ∈ P with c1, c² and c³ as in [\(2.2\)](#page-3-0)-[\(2.3\)](#page-4-0), namely $$p(z) = \frac{1 + (\overline{\tau}_2 \tau_3 + \overline{\tau}_1 \tau_2 + \tau_1)z + (\overline{\tau}_1 \tau_3 + \tau_1 \overline{\tau}_2 \tau_3 + \tau_2)z^2 + \tau_3 z^3}{1 + (\overline{\tau}_2 \tau_3 + \overline{\tau}_1 \tau_2 - \tau_1)z + (\overline{\tau}_1 \tau_3 - \tau_1 \overline{\tau}_2 \tau_3 - \tau_2)z^2 - \tau_3 z^3}, \quad z \in \mathbb{D}.$$

Lemma 2.2. [\[7\]](#page-25-13) Let A, B, C be real numbers and let $$Y(A, B, C) := \max_{z \in \overline{\mathbb{D}}} \{ |A + Bz + Cz^2| + 1 - |z|^2 \}.$$ (i) If AC ≥ 0, then $$Y(A, B, C) = \begin{cases} |A| + |B| + |C|, & \text{if } |B| \ge 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 - |C|)}, & \text{if } |B| < 2(1 - |C|). \end{cases}$$ (ii) If AC < 0, then $$Y(A,B,C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1-|C|)}, & \textit{if} \quad -4AC(C^{-2}-1) \leq B^2 \; \textit{and} \; |B| < 2(1-|C|), \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & \textit{if} \quad B^2 < \min\left\{4(1+|C|)^2, -4AC(C^{-2}-1)\right\}, \\ R(A,B,C), & \textit{otherwise}, \end{cases}$$ where $$R(A,B,C) := \begin{cases} |A| + |B| - |C|, & \text{if } |C|(|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & \text{if } |AB| \le |C|(|B| - 4|A|), \\ (|C| + |A|)\sqrt{1 - \frac{B^2}{4AC}}, & \text{otherwise.} \end{cases}$$

Lemma 2.3. [16] Let $p \in \mathcal{P}$ be given by (2.1). Then $$|c_2 - vc_1^2| \le \begin{cases} -4v + 2, & v < 0, \\ 2, & 0 \le v \le 1, \\ 4v - 2, & v > 1. \end{cases}$$ Moreover, for v < 0 or v > 1, equality holds if and only if $$h(z) = \frac{1+z}{1-z}$$ or one of its rotations. For 0 < v < 1, equality holds if and only if $$h(z) = \frac{1+z^2}{1-z^2}$$ or one of its rotations.

Lemma 2.4. [1] Let $p \in \mathcal{P}$ be given by (2.1) with $0 \le B \le 1$ and $B(2B-1) \le D \le B$ . Then $$\left| c_3 - 2Bc_1c_2 + Dc_1^3 \right| \le 2$$

Lemma 2.5. [21] Let $p \in \mathcal{P}$ be given by (2.1). If $\alpha, \beta, \gamma, \lambda$ satisfy $$0 < \alpha < 1$$ , $0 < \lambda < 1$ , and $$8\lambda(1-\lambda)\Big\{(\alpha\beta-2\gamma)^2+(\alpha(\lambda+\alpha)-\beta)^2\Big\}+\alpha(1-\alpha)(\beta-2\lambda\alpha)^2$$ $$\leq 4\alpha^2(1-\alpha)^2\lambda(1-\lambda),$$ then $$|\gamma c_1^4 + \lambda c_2^2 + 2\alpha c_1 c_3 - \frac{3}{2}\beta c_1^2 c_2 - c_4| \le 2.$$

Lemma 2.6. [23] Let J, K, and L be numbers such that $J \geq 0$ , $K \in \mathbb{C}$ , and $L \in \mathbb{R}$ . Let $p \in \mathcal{P}$ be of the form (2.1) and define a function by $$\Phi(c_1, c_2) = |Kc_1^2 + Lc_2| - |Jc_1|.$$ Then $$\Phi(c_1, c_2) \le \begin{cases} |4K + 2L| - 2J, & \text{if } |2K + L| \ge |L| + J, \\ 2|L|, & \text{otherwise.} \end{cases}$$ and $$-\Phi(c_1, c_2) \le \begin{cases} 2J - M, & when \ J \ge M + 2|L|, \\ 2J\sqrt{\frac{\cdot 2|L|}{M + 2|L|}}, & when \ J^2 \le 2|L|(M + 2|L|), \\ 2|L| + \frac{J^2}{M + 2|L|}, & otherwise, \end{cases}$$ where M = |4K + 2L|.

Theorem 3.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{S}_e^*$ and $\gamma_1, \gamma_2, \gamma_3, \gamma_4$ be given by (1.3). Then we have $$|\gamma_n| \le \frac{1}{2n}$$ , for $n = 1, 2, 3, 4$ . All these bounds are sharp. The following conjecture is proposed for the general coefficients $\gamma_n$ $(n \geq 5)$ of functions in the class $\mathcal{S}_e^*$ . Conjecture 3.1. If $f(z) = z + a_2 z^2 + a_3 z^3 + \dots \in S_e^*$ , then $$|\gamma_n| \le \frac{1}{2n}$$ for $n \in \mathbb{N}$ . The bound is sharp for the functions $f_n$ (for each $n \in \mathbb{N}$ ) defined by (3.1) with $$p(z) = \frac{1+z^n}{1-z^n}.$$

Theorem 3.2. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in C_e$ and $\gamma_1, \gamma_2, \gamma_3, \gamma_4$ be given by (1.3). Then we have $$|\gamma_n| \le \begin{cases} \frac{1}{2n(n+1)}, & n = 1, 2, 3, \\ \frac{1}{8}, & n = 4. \end{cases}$$ All these bounds are sharp.

Theorem 4.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in S_e^*$ . Then $$-\frac{1}{16} \le T_{2,1}(F_f/\gamma) \le \frac{15}{64}.\tag{4.2}$$ Both inequalities in (4.2) are sharp.

Theorem 4.2. Let f(z) = z + a2z ² + a3z ³ + · · · ∈ Ce. Then − 1 144 ≤ T2,1(F^f /γ) ≤ 15 256 . (4.9) Both inequalities in [\(4.9\)](#page-13-1) are sharp Proof. Since f ∈ Ce, substituting the coefficients a² and a³ from [\(3.10\)](#page-8-1)-[\(3.11\)](#page-8-2) into [\(4.1\)](#page-11-4), we obtain $$T_{2,1}(F_f/\gamma) = \frac{1}{36864} \left( -c_1^4 + 576c_1^2 + 32c_1^2 \Re(c_2) - 64|c_2|^2 \right). \tag{4.10}$$ Using Lemma [2.2,](#page-3-0) expression [\(4.10\)](#page-13-2) can be rewritten as $$T_{2,1}(F_f/\gamma) = \frac{1}{36864} \left( -9c_1^4 + 576c_1^2 - 24c_1^2(4 - c_1^2)\Re(\xi) - 16(4 - c_1^2)^2|\xi|^2 \right). \tag{4.11}$$ We first seek the maximum of the right-hand side of [\(4.11\)](#page-13-3). Since −ℜ(ξ) ≤ |ξ|, inequality [\(4.11\)](#page-13-3) gives $$T_{2,1}(F_f/\gamma) \le \frac{1}{36864} \left( -9c_1^4 + 576c_1^2 + 24c_1^2(4 - c_1^2)|\xi| - 16(4 - c_1^2)^2|\xi|^2 \right)$$ $$:= \frac{1}{36864} \Phi(p_1^2, |\xi|). \tag{4.12}$$ If we let p 2 ¹ = x ∈ [0, 4] and |ξ| = y ∈ [0, 1], then $$\Phi(x,y) = -9x^2 + 576x + 24x(4-x)y - 16(4-x)^2y^2. \tag{4.13}$$ Differentiating [\(4.13\)](#page-13-4) partially with respect to x and y, we have $$\Phi_x = -x(18 + 48y + 32y^2) + (576 + 96y + 128y^2),$$ $$\Phi_y = 8(4 - x)(3x - 4(4 - x)y).$$ The system Φ^x = 0 and Φ^y = 0 has no solution within the open region (0, 4) × (0, 1). On the boundary of the closed rectangle [0, 4] × [0, 1], Φ(x, y) takes the following forms: $$\Phi(0,y) = -256y^2$$ and $\Phi(4,y) = 2160$ for all $y \in [0,1]$ , and Φ(x, 0) = −9x ²+576x ≤ 2160 and Φ(x, 1) = −49x ²+800x−256 ≤ 2160 for x ∈ [0, 4]. Hence, $$T_{2,1}(F_f/\gamma) \le \frac{2160}{36864} = \frac{15}{256}.$$ The above inequality is sharp in case of the function f⁵ defined in [\(3.14\)](#page-8-5). We now consider the minimum of [\(4.11\)](#page-13-3). Since ℜ(ξ) ≤ |ξ|, from [\(4.11\)](#page-13-3) it follows that $$T_{2,1}(F_f/\gamma) \ge \frac{1}{36864} \left( -9c_1^4 + 576c_1^2 - 24c_1^2(4 - c_1^2)|\xi| - 16(4 - c_1^2)^2|\xi|^2 \right)$$ $$= \frac{1}{36864} \Psi(p_1^2, |\xi|), \tag{4.14}$$ where $$\Psi(x,y) = -9x^2 + 576x - 24x(4-x)y - 16(4-x)^2y^2. \tag{4.15}$$ Differentiating (4.15) with respect to x and y, we obtain $$\Psi_x = x(-18 + 48y - 32y^2) + 576 - 96y + 128y^2,$$ $$\Psi_y = -8(4 - x)(3x + 4y(4 - x)).$$ The system $\Psi_x = 0$ and $\Psi_y = 0$ has no interior solution in $(0,4) \times (0,1)$ . On the boundary of $[0,4] \times [0,1]$ , the values of $\Psi$ are $$\Psi(0,y) = -256y^2$$ and $\Psi(4,y) = 2160$ for all $y \in [0,1]$ , and $$\Psi(x,0) = -9x^2 + 576x \ge 0$$ and $\Psi(x,1) = -49x^2 + 800x - 256 \ge -256$ for $x \in [0,4]$ . Hence, we have $$T_{2,1}(F_f/\gamma) \ge -\frac{256}{36864} = -\frac{1}{144}.$$ The above inequality is sharp in case of the function $f_6$ defined in (3.15). This completes the

Theorem 5.1. Let $$f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in S_e^*$$ . Then $$|a_2 a_3 - a_4| \le \frac{8}{9\sqrt{7}} \approx 0.336. \tag{5.1}$$ The inequality (5.1) is sharp.

Theorem 5.2. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{C}_e$ . Then $$|a_2 a_3 - a_4| \le \frac{1}{12}. (5.5)$$ The inequality (5) is sharp.

Theorem 6.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in S_e^*$ . Then $$B_1 \le F_{\lambda,\mu}(f) \le \begin{cases} \frac{1}{4} (|3 - 4\lambda| - 4\mu), & \text{if } |3 - 4\lambda| \ge 2 + 4\mu, \\ \frac{1}{2}, & \text{if } |3 - 4\lambda| < 2 + 4\mu. \end{cases}$$ (6.2) where $$B_{1} = \begin{cases} -\frac{1}{4}(4\mu - |3 - 4\lambda|), & \text{if } \frac{\mu+1}{2} \ge |3 - 4\lambda|, \\ -\mu\sqrt{\frac{2}{|3 - 4\lambda| + 2}}, & \text{if } |3 - 4\lambda| \ge \frac{\mu^{2}+1}{2}, \\ -\frac{|3 - 4\lambda| + 16\mu^{2} + 16}{2(|3 - 4\lambda| + 2)}, & \text{if } \frac{\mu+1}{2} < |3 - 4\lambda| < \frac{\mu^{2}+1}{2}. \end{cases}$$ The inequalities in (6.2) are sharp.

Theorem 6.2. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in C_e$ . Then $$B_{1} \leq F_{\lambda,\mu}(f) \leq \begin{cases} \frac{1}{4}(|1-\lambda|-2\mu), & \text{if } |1-\lambda| \geq \frac{2}{3}(2+3\mu), \\ \frac{1}{6}, & \text{if } |1-\lambda| < \frac{2}{3}(2+3\mu). \end{cases}$$ (6.5) where $$B_2 = \begin{cases} -\frac{2\mu - |1 - \lambda|}{4}, & \text{if } \frac{3\mu - 2}{3} \ge |1 - \lambda|, \\ -\frac{1}{2}\mu\sqrt{\frac{2}{3|1 - \lambda| + 2}}, & \text{if } \frac{9\mu^2 - 4}{6} \le |1 - \lambda|, \\ -\frac{9\mu^2 + 6|1 - \lambda| + 4}{12(3|1 - \lambda| + 2)}, & \text{if } \frac{9\mu^2 - 4}{6} > |1 - \lambda| > \frac{3\mu - 2}{3}. \end{cases}$$ The inequalities in (6.5) are sharp.

Definition 1.1. For two analytic functions f and g in a domain $\mathbb{D}$ , we say that f is subordinate to g in $\mathbb{D}$ , and write $f \prec g$ , if there exists a Schwarz function $\omega \in \Omega$ such that $f(z) = g(\omega(z)), z \in \mathbb{D}$ . In particular, if g is univalent in $\mathbb{D}$ , then $f \prec g$ if and only if f(0) = g(0) and $f(\mathbb{D}) \subset g(\mathbb{D})$ . Using the subordination principle, Ma and Minda [16] introduced a unified framework for various subclasses of starlike functions in 1992. They defined $$\mathcal{S}^*(\psi) := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec \psi(z), \ z \in \mathbb{D} \right\},\,$$ and $$C(\psi) := \left\{ f \in \mathcal{S} : 1 + \frac{zf''(z)}{f'(z)} \prec \psi(z), \ z \in \mathbb{D} \right\},\,$$ where $\psi$ is an analytic univalent function with positive real part in $\mathbb{D}$ , symmetric with respect to the real axis, $\psi(0) = 1$ , and $\psi'(0) > 0$ . Interest has grown in studying subclasses of starlike and convex functions for which the superordinate function $\psi(z)$ does not map the entire right half-plane. Although the exponential function is a natural choice for the superordinate function, its selection presents interesting and often non-trivial challenges. The class of starlike functions related to the exponential function $e^z$ , $\mathcal{S}_e$ , was introduced by Mendiratta [15] and is defined by the condition $\frac{zf'(z)}{f(z)} \prec e^z$ . We also recall the related class $\mathcal{C}_e$ og convex functions related to the exponential function, defined by $1 + \frac{zf''(z)}{f'(z)} \prec e^z$ . Precisely, the classes $\mathcal{S}_e$ and $\mathcal{C}_e$ are defined as $$\mathcal{S}_{e}^{*} := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec e^{z}, \ z \in \mathbb{D} \right\},$$ $$\mathcal{C}_{e} := \left\{ f \in \mathcal{S} : 1 + \frac{zf''(z)}{f'(z)} \prec e^{z}, \ z \in \mathbb{D} \right\}.$$ 1.1. Logarithmic coefficients. Note that for $f \in \mathcal{S}$ , let $$F_f(z) := \log \frac{f(z)}{z} = 2 \sum_{n=1}^{\infty} \gamma_n z^n, \quad z \in \mathbb{D}, \quad \log 1 := 0.$$ (1.2) The numbers $\gamma_n := \gamma_n(f)$ are the logarithmic coefficients of f. Few exact upper bounds for $\gamma_n$ exist. These coefficients are known to play a crucial role in the Miliin conjecture ([17], see also [9, p. 155]). Specifically, Miliin [17] 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 established by De Branges [8] in his proof of the Bieberbach conjecture. For the Koebe function $k(z) = \frac{z}{(1-z)^2}$ , the logarithmic coefficients are given by $\gamma_n = \frac{1}{n}$ . Since the Koebe function is the extremal function for many extremal problems in $\mathcal{S}$ , it is natural to conjecture that $|\gamma_n| \leq \frac{1}{n}$ for all $f \in \mathcal{S}$ . However, this conjecture does not hold universally. For example, there exists a bounded function $f \in \mathcal{S}$ with logarithmic coefficients $\gamma_n \neq O(n^{-0.83})$ (see [9, Theorem 8.4]). By differentiating (1.2) and comparing coefficients, the following expressions for $\gamma_n$ in terms of $a_n$ are obtained: $$\begin{cases} \gamma_1 &= \frac{1}{2}a_2, \\ \gamma_2 &= \frac{1}{2}\left(a_3 - \frac{1}{2}a_2^2\right), \\ \gamma_3 &= \frac{1}{2}\left(a_4 - a_2a_3 + \frac{1}{3}a_2^3\right) \\ \gamma_4 &= \frac{1}{2}\left(a_5 - a_2a_4 + a_2^2a_3 - \frac{1}{2}a_3^2 - \frac{1}{4}a_2^4\right). \end{cases}$$ (1.3) Excit forward to show that $|x_1| \le 1$ since $|a_1| \le 2$ . Using the February If $f \in \mathcal{S}$ , it is straightforward to show that $|\gamma_1| \leq 1$ , since $|a_2| \leq 2$ . Using the Fekete-Szegö inequality (see [9, Theorem 3.8]) for functions in $\mathcal{S}$ and substituting into (1.2), the sharp estimate for $\gamma_2$ is given by $$|\gamma_2| \le \frac{1}{2}(1 + 2e^{-2}) \approx 0.635.$$ For $n \geq 3$ , deriving bounds for $|\gamma_n|$ is considerably more challenging, and no significant general bounds for $|\gamma_n|$ for functions in S are currently known. Logarithmic coefficients have recently been a focus of research interest for various authors (e.g., [2, 3, 5, 10, 12, 18, 19]). In this article, we investigate various coefficient problems and determine their sharp bounds for several topics in geometric function theory, specifically focusing on the logarithmic coefficients, Hermitian-Toeplitz determinant, generalized Zalcman conjecture, and the generalized Fekete-Szegö inequality. The remainder of the paper is organized as follows: Section 2 introduces the necessary lemmas required to establish our main findings. Section 3 establishes sharp bounds for the logarithmic coefficients of the classes $S_e$ and $C_e$ . Section 4 presents sharp bounds of the Second-order Hermitian-Toeplitz determinant of logarithmic coefficients for the classes $S_e$ and $C_e$ . In Section 5, the generalized Zalcman conjecture for the classes $S_e$ and $C_e$ is discussed. Finally, Section 6 establishes sharp bounds of the generalized Fekete-Szegö functional for the classes $S_e$ and $C_e$ . The proofs of the main results are discussed in detail in each respective section.