Sharp coefficient Estimates for the class \(\mathcal{S}_{\mathcal{AP}}^{*}\)
January 01, 1970
Abstract
We investigate several classic coefficient problems for the Ma–Minda starlike subclass \(\mathcal{S}_{\mathcal{AP}}^{*}\) defined by the apple-like subordination function \(\psi_{\mathcal{AP}}(z)=e^{z}\sqrt{1+z}\). Sharp bounds are derived for the initial inverse logarithmic coefficients \(\Gamma_1\), \(\Gamma_2\), \(\Gamma_3\), and the successive modulus difference \(|\Gamma_2|-|\Gamma_1|\). In addition, we evaluate the second-order inverse logarithmic Hankel determinant, the generalized Fekete–Szegö functional over all real parameter domains, and the third-order Hermitian–Toeplitz determinant. The corresponding extremal functions are explicitly determined for each functional.
1 Introduction↩︎
Let \(\mathcal{H}\) denote the class of analytic functions defined in the open unit disk \(\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}\), equipped with the topology of uniform convergence on compact subsets. We denote by \(\mathcal{A}\) the subclass of \(\mathcal{H}\) consisting of functions normalized by \(f(0)=0\) and \(f'(0)=1\). Let \(\mathcal{S}\) be the family of functions \(f\in\mathcal{A}\) that are univalent in \(\mathbb{D}\). Every function \(f\in\mathcal{S}\) can be expressed by the Taylor series expansion \[\label{eq1} f(z)=z+\sum_{n=2}^{\infty}a_nz^n, \quad z\in\mathbb{D}.\tag{1}\]
A function \(f\in\mathcal{A}\) is starlike if the image domain \(f(\mathbb{D})\) is starlike with respect to the origin, and is convex if \(f(\mathbb{D})\) is a convex domain. These standard subclasses of \(\mathcal{S}\) are denoted by \(\mathcal{S}^{\ast}\) and \(\mathcal{C}\), respectively. Analytically, a function \(f\in\mathcal{A}\) belongs to \(\mathcal{S}^{\ast}\) if and only if \(\Re\left(zf'(z)/f(z)\right)>0\) for all \(z\in\mathbb{D}\), and belongs to \(\mathcal{C}\) if and only if \(\Re\left(1+zf''(z)/f'(z)\right)>0\) for all \(z\in\mathbb{D}\). The classical Alexander relation states that \(f\in\mathcal{C}\) if and only if \(zf'\in\mathcal{S}^{\ast}\).
Recall that an analytic function \(f\) is subordinate to an analytic function \(g\) in \(\mathbb{D}\), written as \(f\prec g\), if there exists a Schwarz function \(\omega\) (analytic in \(\mathbb{D}\) with \(\omega(0)=0\) and \(|\omega(z)|<1\)) such that \(f(z)=g(\omega(z))\) for \(z\in\mathbb{D}\). In particular, if \(g\) is univalent in \(\mathbb{D}\), then \(f\prec g\) is equivalent to \(f(0)=g(0)\) and \(f(\mathbb{D})\subset g(\mathbb{D})\).
Using the principle of subordination, Ma and Minda [1] introduced a unified framework for subclasses of starlike functions, defined by \[\mathcal{S}^{\ast}(\psi) = \left\{ f\in\mathcal{S}: \frac{zf'(z)}{f(z)} \prec \psi(z), \quad z\in\mathbb{D}\right\},\] where \(\psi\) is an analytic and univalent function mapping \(\mathbb{D}\) onto a domain that is symmetric with respect to the real axis, with \(\Re(\psi(z))>0\), \(\psi(0)=1\), and \(\psi'(0)>0\).
Various choices for the superordinate function \(\psi\) have been studied extensively [2]–[9] to model particular geometric configurations and coefficient behavior. Recently, Sana, Saliu, and Riaz [10] introduced the subclass \(\mathcal{S}_{\mathcal{AP}}^{\ast}\) associated with the apple-like generating function \(\psi_{\mathcal{AP}}(z)=e^{z}\sqrt{1+z}\). A function \(f\in\mathcal{A}\) belongs to this class if \[\frac{zf'(z)}{f(z)}\prec\psi_{\mathcal{AP}}(z), \quad z\in\mathbb{D}.\] The function \(\psi_{\mathcal{AP}}\) maps the unit disk onto a symmetric, apple-like region and satisfies the standard Ma–Minda conditions, positioning \(\mathcal{S}_{\mathcal{AP}}^{\ast}\) as a structural subclass of the broader Ma–Minda starlike family.
The primary objective of this paper is to investigate several extremal problems within the class \(\mathcal{S}_{\mathcal{AP}}^{\ast}\). Specifically, we study its inverse logarithmic coefficients, consecutive coefficient differences, and structural determinants to map the geometric bounds imposed by this target domain.
1.1 Inverse Logarithmic Coefficients↩︎
For a function \(f\in\mathcal{S}\), the inverse logarithmic coefficients \(\Gamma_n\) are defined by Ponnusamy et al. [11] via the log-expansion of the inverse function \(f^{-1}\): \[F_{f^{-1}}(w):= \log\frac{f^{-1}(w)}{w} = 2\sum_{n=1}^{\infty}\Gamma_n w^n, \quad |w|<\frac{1}{4}.\] By explicit computation, the first four coefficients are related to the Taylor coefficients of \(f\) by \[\label{IG1} \begin{cases} \Gamma_1 = -\frac{1}{2} a_2, \\ \Gamma_2 = -\frac{1}{2} a_3 + \frac{3}{4} a_2^2, \\ \Gamma_3 = -\frac{1}{2} \left( a_4 - 4a_2 a_3 + \frac{10}{3} a_2^3 \right), \\ \Gamma_4 = \frac{35}{8} a_2^4 - \frac{15}{2} a_2^2 a_3 + \frac{5}{2}a_2 a_4 + \frac{5}{4} a_3^2 - \frac{1}{2}a_5. \end{cases}\tag{2}\] Ponnusamy et al. [11] proved that for the full class \(\mathcal{S}\), the sharp inequality \(|\Gamma_n| \le \frac{1}{2n}\binom{2n}{n}\) holds for all \(n\in\mathbb{N}\), with equality if and only if \(f\) is the Koebe function or one of its rotations.
Determinants whose entries are logarithmic coefficients provide useful structural properties of univalent functions. Kowalczyk and Lecko [12], [13] studied the classical Hankel determinant \(H_{q,n}(F_f/2)\) for logarithmic coefficients. Its inverse analogue, \(H_{q,n}(F_{f^{-1}}/2)\), utilizes the inverse logarithmic coefficients \(\Gamma_n\) as entries (see [14], [15]): \[H_{q,n}\left(F_{f^{-1}}/2\right) = \begin{vmatrix} \Gamma_n & \Gamma_{n+1} & \cdots & \Gamma_{n+q-1} \\ \Gamma_{n+1} & \Gamma_{n+2} & \cdots & \Gamma_{n+q} \\ \vdots & \vdots & \ddots & \vdots \\ \Gamma_{n+q-1} & \Gamma_{n+q} & \cdots & \Gamma_{n+2(q-1)} \end{vmatrix}.\]
While the standard Taylor and logarithmic coefficients for functions associated with the apple-like domain have recently been examined by Sana et al. [10], several corresponding problems for the inverse configurations remain open. To bridge this gap, this paper establishes the following sharp results for the class \(\mathcal{S}_{\mathcal{AP}}^{\ast}\):
Upper bounds for the initial inverse logarithmic coefficients \(\Gamma_1\), \(\Gamma_2\), and \(\Gamma_3\).
Estimates for the second-order inverse logarithmic Hankel determinant \(H_{2,1}(F_{f^{-1}}/2) = \Gamma_1\Gamma_3-\Gamma_2^2\).
Upper and lower bounds for the consecutive coefficient difference \(|\Gamma_2|-|\Gamma_1|\).
Sharp bounds for the generalized Fekete–Szegö functional \(|a_3 - \lambda a_2^2| - \mu |a_2|\) over all real parameters \(\lambda\) and \(\mu > 0\).
Sharp bounds for the third-order Hermitian–Toeplitz determinant \(T_{3,1}(f)\).
2 Auxiliary lemmas↩︎
Let \(\mathcal{P}\) denote the class of all analytic functions \(p\) in the unit disk \(\mathbb{D}\) satisfying \(p(0) = 1\) and \(\Re p(z) > 0\) for all \(z \in \mathbb{D}\). Then, every \(p \in \mathcal{P}\) admits the series representation \[\label{p1} p(z) = 1 + \sum_{n=1}^{\infty} c_n z^n, \quad z \in \mathbb{D}.\tag{3}\] Functions in \(\mathcal{P}\) are referred to as Carathéodory functions. It is well-known that for \(p \in \mathcal{P}\), the coefficients satisfy the sharp bound \(|c_n| \leq 2\) for all \(n \geq 1\) (see [16]). The Carathéodory class \(\mathcal{P}\) and its coefficient bounds play a fundamental role in deriving estimates for sharp bounds in geometric function theory.
Now, we state some lemmas, which will be useful to establish our main results: Now we recall the following well-known result due to Cho et al. [17].
Lemma 1. [17] If \(p\in\mathcal{P}\) is of the form (3 ), then \[\label{c1}c_1 =2\tau_1,\qquad{(1)}\] \[\label{c2} c_2=2\tau_1^2 + 2(1 - |\tau_1|^2)\tau_2\qquad{(2)}\] and \[\label{c3} c_3 = 2\tau_1^3+4(1-|\tau_1|^2)\tau_1\tau_2 - 2(1 - |\tau_1|^2)\overline{\tau_1}\tau_2^2 + 2(1 - \tau_1^2)(1 - |\tau_2|^2)\tau_3\qquad{(3)}\] for some \(\tau_1, \tau_2, \tau_3 \in\mathbb{\overline{D}}:= \{z \in \mathbb{C}: |z| \leq 1 \}\). For \(\tau_1 \in \mathbb{T} := \{ z \in \mathbb{C} : |z| = 1 \}\) , there is a unique function \(p \in \mathcal{P}\) with \(c_1\) as in (?? ), namely, \[p(z) = \frac{1 + \tau_1 z}{1 - \tau_1 z}, \quad z \in \mathbb{D}.\] For \(\tau_1 \in \mathbb{D}\) and \(\tau_2 \in \mathbb{T}\) , there is a unique function \(p \in \mathcal{P}\) with \(c_1\) and \(c_2\) as in (?? ) and (?? ), 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 \(\tau_1, \tau_2 \in \mathbb{D}\) and \(\tau_3 \in \mathbb{T}\) , there is a unique function \(p \in \mathcal{P}\) with \(c_1\) , \(c_2\) , and \(c_3\) as in (?? ?? ), namely, \[p(z) = \frac{1 + (\overline{\tau}_2 \tau_3 + \overline{\tau}_1 \tau_2 + \tau_1)z+(\overline{\tau}_1\tau_3+\tau_1\overline{\tau}_2\tau_3+\tau_2)z^2+\tau_3z^3}{1+(\overline{\tau}_2\tau_3+\overline{\tau}_1\tau_2-\tau_1)z+(\overline{\tau}_1\tau_3-\tau_1\overline{\tau}_2\tau_3-\tau_2)z^2-\tau_3z^3},\;\;z\in\mathbb{D}\]
Following well-known result is due to Choi et al. [18].
Lemma 2. [18] Let \(A\), \(B\), \(C\) be real numbers and let \[Y(A, B, C):= \max\limits_{z\in \overline{\mathbb{D}}}\left\lbrace |A+Bz+Cz^2|+1-|z|^2\right\rbrace.\]
If \(AC\geq 0\), then \[Y(A, B, C) = \begin{cases} |A|+|B|+|C|, & \text{if}\;\;\; |B|\geq 2(1-|C|), \\ 1+|A|+\frac{B^2}{4(1-|C|)}, &\text{if}\;\;\; |B|<2(1-|C|). \end{cases}\]
If \(AC<0\), then \[Y(A,B,C)= \begin{cases} 1-|A|+\frac{B^2}{4(1-|C|)}, &\text{if}\;\;\; -4AC(C^{-2}-1) \leq B^2\; \text{and}\; |B|<2(1-|C|), \\ 1+|A|+\frac{B^2}{4(1+|C|)}, &\text{if}\;\;\; B^2<\min\left\{4(1+|C|)^2, -4AC(C^{-2}-1) \right\}, \\ R(A,B,C), &\text{otherwise}, \end{cases}\] where \[R(A,B,C):= \begin{cases} |A|+|B|-|C|, & \text{if}\;\;\; |C|(|B|+4|A|) \leq |AB|, \\ -|A|+|B|+|C|, & \text{if}\;\;\; |AB|\leq |C|(|B|-4|A|), \\ (|C|+|A| )\sqrt{1-\frac{B^2}{4AC}}, &\text{otherwise}. \end{cases}\]
Lemma 3. [1] Let \(p \in \mathcal{P}\) be given by 3 . Then \[\left| c_2 - v c_1^2 \right| \le \begin{cases} -4v + 2, & v < 0, \\ 2, & 0 \leq v \leq 1, \\ 4v - 2, & v > 1. \end{cases}\] Moreover, for \(v < 0\) or \(v > 1\), equality holds if and only if \[h(z) = \frac{1+z}{1-z} \quad \text{or one of its rotations}.\] For \(0 < v < 1\), equality holds if and only if \[h(z) = \frac{1+z^2}{1-z^2} \quad \text{or one of its rotations}.\]
Lemma 4. [19] 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 (3 ) and define a function by \[\Phi(c_1,c_2) = \big| K c_1^2 + L c_2 \big| - \big| J c_1 \big|.\] Then \[\Phi(c_{1}, c_{2}) \le \begin{cases} |4K + 2L| - 2J, & \text{if } |2K + L| \geq |L| + J, \\[6pt] 2|L|, & \text{otherwise.} \end{cases}\] and \[-\Phi(c_1,c_2) \leq \begin{cases} 2J - M, & \text{when } J \geq M + 2|L|, \\[6pt] 2J \sqrt{\dfrac{ 2|L|}{M + 2|L|}}, & \text{when } J^2 \leq 2|L|(M + 2|L|), \\[10pt] 2|L| +\dfrac{ J^2}{M + 2|L|}, & \text{otherwise} \end{cases}\] where \(M=|4K+2L|\).
3 Sharp bounds for the logarithmic inverse coefficients of the class \(\mathcal{S}_{\mathcal{AP}}^{*}\).↩︎
Theorem 1. Let \(f \in \mathcal{S}_{\mathcal{AP}}^{*}\), and let the inverse logarithmic coefficients \(\Gamma_n\) (\(n \geq 1\)) be defined by 2 . Then \[|\Gamma_1| \le \frac{3}{4}, \qquad |\Gamma_2| \le \frac{29}{32}, \qquad |\Gamma_3| \le \frac{925}{576}.\] All these estimates are sharp.
Proof. Since \(f \in \mathcal{S}_{\mathcal{AP}}^{*}\), there exists a Schwarz function \(w(z) = \frac{p(z)-1}{p(z)+1}\) with \(p \in \mathcal{P}\) such that \(\frac{zf'(z)}{f(z)} = e^{w(z)}\sqrt{1+w(z)}\). Expanding \(w(z)\) in powers of \(z\) using the coefficients \(c_n\) of \(p\) yields \[w(z) = \frac{c_1}{2}z + \left( \frac{c_2}{2} - \frac{c_1^2}{4} \right)z^2 + \left( \frac{c_3}{2} - \frac{c_1c_2}{2} + \frac{c_1^3}{8} \right)z^3 + \cdots\] Substituting this into the power series expansion of the subordination function gives \[e^{w(z)}\sqrt{1+w(z)} = 1 + \frac{3}{2}w(z) + \frac{7}{8}w^2(z) + \frac{17}{48}w^3(z) + \frac{19}{128}w^4(z) + \cdots\] Equating coefficients in \(zf'(z) = f(z)\left(1 + \sum_{n=1}^{\infty} q_n z^n\right)\), we obtain the Taylor coefficients of \(f\): \[\label{pn1} \begin{align} a_2 &= \frac{3}{4}c_1, \\ a_3 &= \frac{3}{8}c_2 + \frac{13}{64}c_1^2, \\ a_4 &= \frac{1}{4}c_3 + \frac{17}{96}c_1c_2 + \frac{37}{2304}c_1^3. \end{align}\tag{4}\]
By the definition \(\Gamma_1 = -\frac{1}{2}a_2\) and 4 , we have \[|\Gamma_1| = \left|-\frac{1}{2}\cdot \frac{3}{4}c_1\right| = \frac{3}{8}|c_1| \le \frac{3}{4},\] since \(|c_1| \le 2\) for all \(p \in \mathcal{P}\). Equality holds if and only if \(|c_1|=2\), which corresponds to the function \[\label{pn2} f_0(z) = z\exp\left( \int_0^z \frac{e^t\sqrt{1+t}-1}{t}\,dt \right) = z + \frac{3}{2}z^2 + \frac{25}{16}z^3 + \frac{385}{288}z^4 + \cdots.\tag{5}\]
From 2 and 4 , the second coefficient satisfies \[|\Gamma_2| = \left| -\frac{1}{2} a_3 + \frac{3}{4} a_2^2 \right| = \left| -\frac{1}{2}\left( \frac{3}{8} c_2 + \frac{13}{64}c_1^2 \right) + \frac{3}{4}\left( \frac{3}{4} c_1 \right)^2 \right| = \frac{3}{16} \left| c_2 - \frac{41}{24}c_1^2 \right|.\] Applying Lemma 3 with \(v = \frac{41}{24} > 1\), we get \(\left| c_2 - \frac{41}{24}c_1^2 \right| \le 4\left(\frac{41}{24}\right)-2 = \frac{29}{6}\). Therefore, \(|\Gamma_2| \le \frac{3}{16} \cdot \frac{29}{6} = \frac{29}{32}\), which is attained by \(f_0\).
To estimate \(\Gamma_3\), we substitute the relations from 4 into the definition of \(\Gamma_3\), which yields \[\label{G3} |\Gamma_3| = \left| -\frac{1}{2} \left( a_4 - 4 a_2 a_3 + \frac{10}{3} a_2^3 \right) \right| = \frac{1}{8} \left| c_3 - \frac{91}{24} c_1 c_2 + \frac{1873}{576} c_1^3 \right|.\tag{6}\] Expressing the coefficients \(c_n\) in terms of the parameters \(\tau_1, \tau_2, \tau_3 \in \overline{\mathbb{D}}\) using Lemma 1 converts 6 into \[\label{G11} |\Gamma_3| = \frac{1}{8} \left| \frac{925}{72}\tau_1^3 - \frac{67}{6}\tau_1(1 - |\tau_1|^2)\tau_2 - 2\overline{\tau_1}(1 - |\tau_1|^2)\tau_2^2 + 2(1 - |\tau_1|^2)(1 - |\tau_2|^2)\tau_3 \right|.\tag{7}\] Because the class \(\mathcal{P}\) and the functional are rotationally invariant, we assume without loss of generality that \(c_1 \in [0,2]\), so that \(\tau_1 \in [0,1]\). We consider two cases for \(\tau_1\):
Case 1. If \(\tau_1 = 1\), the terms containing \((1-|\tau_1|^2)\) vanish identically in 7 , which immediately gives \[|\Gamma_3| = \frac{925}{576}.\]
Case 2. If \(0 \le \tau_1 < 1\), let \(x = \tau_1 \in [0,1)\). Applying the triangle inequality to 7 yields \[\label{G12} |\Gamma_3| \le \frac{1 - x^2}{4} \left( \left| A + B \tau_2 + C \tau_2^2 \right| + 1 - |\tau_2|^2 \right),\tag{8}\] where \[\label{pn3} A = \frac{925x^3}{144(1 - x^2)} > 0, \qquad B = -\frac{67}{12}x, \qquad C = -x.\tag{9}\] Since \(AC < 0\), we apply part (ii) of Lemma 2:
Case 2(a). If \(-4AC(C^{-2}-1) \le B^2\) and \(|B| < 2(1-|C|)\), the second condition simplifies to \(\frac{67}{12}x < 2(1-x)\), which gives \(x \in [0, \frac{24}{91})\). Since \(-4AC(C^{-2}-1) = \frac{925}{36}x^2\) and \(B^2 = \frac{4489}{144}x^2\), the inequality \(\frac{925}{36} < \frac{4489}{144}\) holds on this interval. Lemma 2 gives the maximum value as \(1 - |A| + \frac{B^2}{4(1-|C|)}\). Substituting this into 8 yields \[|\Gamma_3| \le \frac{1-x^2}{4} \left( 1 - \frac{925x^3}{144(1-x^2)} + \frac{4489x^2}{576(1-x)} \right).\] Simplifying this expression gives the polynomial \[G(x) = \frac{789x^3 + 3913x^2 + 576}{2304}.\] Since \(G'(x) = \frac{2367x^2 + 7826x}{2304} > 0\) for all \(x > 0\), the function \(G\) is strictly increasing on \([0, \frac{24}{91})\), and its maximum value is \[G\left( \frac{24}{91} \right) \approx 0.374 < \frac{925}{576}.\]
Case 2(b). If \(B^2 < \min\left\{4(1+|C|)^2, -4AC(C^{-2}-1)\right\}\), the condition requires \(B^2 < -4AC(C^{-2}-1)\). Substituting the parameters gives \(\frac{4489}{144}x^2 < \frac{3700}{144}x^2\), which implies \(4489 < 3700\). This contradiction shows that this case cannot occur for \(x \in (0,1)\).
Case 2(c). The condition \(|C|(|B|+4|A|) \le |AB|\) simplifies to \(x\left(\frac{67}{12}x + \frac{925x^3}{36(1-x^2)}\right) \le \frac{61975x^4}{1728(1-x^2)}\), which reduces to \(9648x^2 + 17575x^4 \le 0\). This inequality does not hold for any \(x \in (0,1)\).
Case 2(d). If \(|AB| \le |C|(|B|-4|A|)\), the condition simplifies to \(9648x^2 - 27223x^4 \le 0\), which restricts \(x\) to the interval \(x \ge \sqrt{\frac{9648}{27223}} \approx 0.5954\). Here, Lemma 2 gives the maximum value as \(|A| + |B| - |C|\). Substituting these parameters into 8 removes the denominator \(1-x^2\), leaving \[|\Gamma_3| \le \frac{1-x^2}{4}\left( \frac{925x^3}{144(1-x^2)} + \frac{67}{12}x - x \right) = \frac{265x^3 + 660x}{576} =: \Phi(x).\] Since \(\Phi'(x) = \frac{795x^2 + 660}{576} > 0\), the function \(\Phi\) is strictly increasing on this interval and reaches its upper bound at the endpoint: \[\lim_{x \to 1^-} \Phi(x) = \frac{265 + 660}{576} = \frac{925}{576}.\]
Case 2(e). For the remaining case where the maximum value is given by \((|C|+|A| ) \sqrt{1-\frac{B^2}{4AC}}\), substituting the parameters from 9 into 8 gives \[|\Gamma_3| \le \frac{144+781x^2}{576} \sqrt{\frac{4489-789x^2}{3700}} =: \Omega(x).\] To show that \(\Omega(x) < \frac{925}{576}\) on \((0,1)\), we consider the equivalent inequality \((144+781x^2)^2(4489-789x^2) < 925^2 \cdot 3700\). Let \(t = x^2 \in (0,1)\) and define \(F(t) = 925^2 \cdot 3700 - (144+781t)^2(4489-789t)\). Note that \(F(1) = 0\). Differentiating \(F(t)\) yields \[F'(t) = (781t+144)(1848627t-6898202).\] Because \(781t+144 > 0\) and \(1848627t-6898202 < 0\) for all \(t \in (0,1)\), it follows that \(F'(t) < 0\). Thus, \(F(t)\) decreases strictly to its minimum value of \(0\) at \(t=1\), meaning \(F(t) > 0\) for all \(t \in (0,1)\). This proves that \(\Omega(x) < \frac{925}{576}\).
As a result, the maximum value is determined by Case 1, establishing that \(|\Gamma_3| \le \frac{925}{576}\). ◻
3.1 Sharpness and Geometric Extremality↩︎
To verify the sharpness of the bounds obtained in Theorem [T1], we examine the behavior of the rotationally invariant function \(f_0 \in \mathcal{S}_{\mathcal{AP}}^{*}\) defined in 5 , which satisfies the structural differential relation \[\label{pn4} \frac{zf_0'(z)}{f_0(z)} = e^z \sqrt{1+z}.\tag{10}\] The analytic mapping \(\psi_{\mathcal{AP}}(z) = e^z \sqrt{1+z}\) maps the open unit disk \(\mathbb{D}\) onto a bounded, slit-free apple-like domain symmetric with respect to the real axis. Integrating the power series expansion of 10 gives the initial Taylor coefficients of \(f_0(z)\): \[a_2 = \frac{3}{2}, \qquad a_3 = \frac{25}{16}, \qquad a_4 = \frac{385}{288}, \qquad a_5 = \frac{383}{384}.\] Substituting these values directly into the definitions of the inverse logarithmic coefficients \(\Gamma_n\) given in 2 yields:
For \(\Gamma_1\): \[\Gamma_1 = -\frac{1}{2}a_2 = -\frac{3}{4} \implies |\Gamma_1| = \frac{3}{4}.\]
For \(\Gamma_2\): \[\Gamma_2 = -\frac{1}{2}a_3 + \frac{3}{4}a_2^2 = -\frac{25}{32} + \frac{27}{16} = \frac{29}{32} \implies |\Gamma_2| = \frac{29}{32}.\]
For \(\Gamma_3\): \[\Gamma_3 = -\frac{1}{2}\left(a_4 - 4a_2 a_3 + \frac{10}{3}a_2^3\right) = -\frac{1}{2}\left(\frac{385}{288} - \frac{75}{8} + \frac{45}{4}\right) = -\frac{925}{576} \implies |\Gamma_3| = \frac{925}{576}.\]
In each case, the absolute value of the coefficient matches the upper bound precisely, proving sharpness. The conformal deformation from the unit disk \(\mathbb{D}\) to the univalent image domain \(f_0(\mathbb{D})\) is shown in Figure 2.
4 Bounds for the differences of logarithmic inverse coefficients for \(\mathcal{S}_{\mathcal{AP}}^{*}\)↩︎
The Bieberbach conjecture, proved by de Branges [20], states that the Taylor coefficients of any function \(f \in \mathcal{S}\) of the form 1 satisfy the bound \(|a_n| \leq n\) for all \(n \geq 2\), with equality holding only for the Koebe function and its rotations. This result motivated extensive study into the behavior of differences of consecutive coefficients. For instance, the inequality \[\bigl||a_{n+1}| - |a_n|\bigr| \leq 1, \quad n \geq 2,\] was shown to hold for starlike functions by Leung [21], following a conjecture by Pommerenke [22]. For convex functions, similar coefficient differences were investigated by Li and Sugawa [23].
Recently, attention has focused on the differences of logarithmic coefficients \(|\gamma_{n+1}| - |\gamma_n|\) (see Lecko and Partyka [24], Obradović and Tuneski [25], and Kumar and Cho [26]). In this section, we determine sharp upper and lower bounds for the difference of the initial inverse logarithmic coefficients, \(|\Gamma_2| - |\Gamma_1|\), for functions in the class \(\mathcal{S}_{\mathcal{AP}}^{*}\).
Theorem 2. Let \(f \in \mathcal{S}_{\mathcal{AP}}^{*}\) and let the inverse logarithmic coefficients \(\Gamma_n\) (\(n = 1,2\)) be defined by 2 . Then \[\label{T2} -\frac{3\sqrt{3}}{2\sqrt{41}} \le |\Gamma_2|-|\Gamma_1| \le \frac{3}{8}.\qquad{(4)}\] The inequalities are sharp.
Proof. Let \(f \in \mathcal{S}_{\mathcal{AP}}^{*}\). From the relation between the inverse logarithmic coefficients and the Carathéodory coefficients, we have \[\label{pn6} |\Gamma_2|-|\Gamma_1| = \frac{3}{16}\left|c_2-\frac{41}{24}c_1^2\right| -\frac{3}{8}|c_1| = \frac{1}{128}\left(|41c_1^2-24c_2|-48|c_1|\right) = \frac{1}{128}\Phi(c_1,c_2),\tag{11}\] where \(\Phi(c_1,c_2) = |Kc_1^2+Lc_2|-|Jc_1|\) with \(K=41\), \(L=-24\), and \(J=48\).
Calculating the auxiliary parameters for Lemma 4 gives \(M = |4K+2L| = 116\). Since \(|2K+L| = 58 < 72 = |L|+J\), the upper bound condition yields \(\Phi(c_1,c_2) \le 2|L| = 48\). For the lower bound, because \(J^2 = 2304 < 2|L|(M+2|L|) = 7872\), Lemma 4 gives \(-\Phi(c_1,c_2) \le 2J \sqrt{\frac{2|L|}{M+2|L|}} = \frac{192\sqrt{3}}{\sqrt{41}}\). Substituting these bounds into 11 gives the inequalities stated in Theorem ?? . ◻
4.1 Sharpness and Extremal Domains↩︎
To establish the sharpness of the upper bound in Theorem ?? , we consider the function \(f_1 \in \mathcal{S}_{\mathcal{AP}}^{*}\) defined by \[\label{mn7} f_1(z) = z\exp\left( \int_0^z \frac{e^{t^2}\sqrt{1+t^2}-1}{t}\,dt \right).\tag{12}\] This maps to the Schwarz function \(\omega(z)=z^2\), which corresponds to \(p(z) = \frac{1+z^2}{1-z^2} = 1+2z^2+2z^4+\cdots\). Thus, \(c_1=0\) and \(c_2=2\). This yields \(a_2=0\) and \(a_3=\frac{3}{4}\), which implies \(\Gamma_1 = 0\) and \(\Gamma_2 = -\frac{3}{8}\). Consequently, \(|\Gamma_2|-|\Gamma_1| = \frac{3}{8}\). The geometric structure of the two-fold symmetric domain \(f_1(\mathbb{D})\) is shown in Figure 3.
To establish the sharpness of the lower bound in ?? , we consider the function \(f_2 \in \mathcal{S}_{\mathcal{AP}}^{*}\) defined by \[\label{mn8} f_2(z) = z\exp\left( \int_0^z \frac{e^{\omega(t)}\sqrt{1+\omega(t)}-1}{t}\,dt \right),\tag{13}\] where \(\omega(z) = z\left(z+\frac{2\sqrt{3}}{\sqrt{41}}\right) / \left(1+\frac{2\sqrt{3}}{\sqrt{41}}z\right)\). The coefficients of the corresponding function \(p(z) = \frac{1+\omega(z)}{1-\omega(z)}\) are \(c_1 = \frac{4\sqrt{3}}{\sqrt{41}}\) and \(c_2=2\). This gives \(\Gamma_1 = -\frac{3\sqrt{3}}{2\sqrt{41}}\) and \(\Gamma_2 = 0\), which yields \(|\Gamma_2|-|\Gamma_1| = -\frac{3\sqrt{3}}{2\sqrt{41}}\). The asymmetric layout of the domain \(f_2(\mathbb{D})\) is shown in Figure 4.
5 Hankel determinants for the inverse logarithmic coefficients for \(\mathcal{S}_{\mathcal{AP}}^{*}\)↩︎
Theorem 3. Let \(f\in \mathcal{S}_{\mathcal{AP}}^{*}\) be given by 1 . Then \[\left| H_{2,1}\left(F_{f^{-1}} / 2\right)\right| \le \frac{1177}{3072}.\] The inequality is sharp.
Proof. Substituting the inverse logarithmic transformations from 2 and the corresponding Taylor coefficients from 4 into the definition of the second-order Hankel determinant yields \[\label{mn9} H_{2,1}(F_{f^{-1}} / 2) = \Gamma_1 \Gamma_3 - \Gamma_2^2 = \frac{2449}{49152} c_1^4 - \frac{59}{1024} c_1^2 c_2 + \frac{3}{64} c_1 c_3 - \frac{9}{256} c_2^2.\tag{14}\] By applying Lemma 1 to express the coefficients \(c_n\) in terms of \(\tau_n \in \overline{\mathbb{D}}\), 14 expands to \[\label{mn10} |H_{2,1}(F_{f^{-1}} / 2)| = \left| \frac{1177}{3072}\tau_1^4 - \frac{1 - |\tau_1|^2}{128} \left( 47\tau_1^2\tau_2 + 24|\tau_1|^2\tau_2^2 - 24\tau_1(1-|\tau_2|^2)\tau_3 + 18(1-|\tau_1|^2)\tau_2^2 \right) \right|.\tag{15}\] Since the class \(\mathcal{P}\) and the functional \(H_{2,1}(F_{f^{-1}}/2)\) are invariant under rotational transformations, we assume without loss of generality that \(c_1 \in [0,2]\), which implies \(\tau_1 \in [0,1]\). We analyze the functional under three cases determined by the value of \(\tau_1\):
Case 1. If \(\tau_1=0\), then 15 simplifies to \[\left|H_{2,1}(F_{f^{-1}}/2)\right| \le \frac{18}{128}|\tau_2|^2 \le \frac{9}{64} = 0.140625.\]
Case 2. If \(\tau_1=1\), the term containing \((1-|\tau_1|^2)\) in 15 vanishes identically, leaving \[\left|H_{2,1}(F_{f^{-1}}/2)\right| = \frac{1177}{3072} \approx 0.383139.\]
Case 3. If \(\tau_1 \in (0,1)\), let \(x = \tau_1 \in (0,1)\). Applying the triangle inequality to 15 yields \[\label{mn11} \left| H_{2,1}(F_{f^{-1}} / 2) \right| \le \frac{3x (1 - x^2)}{16} \left( |A + B \tau_2 + C \tau_2^2| + 1 - |\tau_2|^2 \right),\tag{16}\] where \[A = \frac{1177x^3}{576(1-x^2)}, \qquad B=-\frac{47}{24}x, \qquad C=-\frac{3+x^2}{4x}.\] Since \(A > 0\) and \(C < 0\), we have \(AC < 0\). We evaluate the conditions from part (ii) of Lemma 2:
Case 3(a). The condition \(|B|<2(1-|C|)\) cannot be satisfied because \(|C| = \frac{3+x^2}{4x} > 1\) for all \(x \in (0,1)\), which implies \(2(1-|C|)<0\), while \(|B| > 0\).
Case 3(b). The condition \(B^2 < -4AC(C^{-2}-1)\) is not satisfied because \(|C|>1\) implies \(C^{-2}-1<0\), which makes the right-hand side negative while \(B^2 \ge 0\).
Case 3(c). The condition \(|C|(|B|+4|A|)\le |AB|\) simplifies to \(33839x^4-71208x^2-20304\ge0\). Setting \(t=x^2\), the unique positive root of this quadratic is \(t_k \approx 2.3536\). Since \(t_k > 1\), this inequality does not hold for \(0<x<1\).
Case 3(d). The condition \(|AB|\le |C|(|B|-4|A|)\) simplifies to the polynomial inequality \(33839x^4+31224x^2-20304\le0\). This defines the sub-interval \(0<x\le x_m \approx 0.6360\). In this region, Lemma 2 sets \(Y(A,B,C) = -|A|+|B|+|C|\). Substituting this into 16 shows that the maximum value on this interval is bounded above by \(0.15938\), which is strictly less than \(\frac{1177}{3072}\).
Case 3(e). In the remaining sub-case where \(Y(A,B,C) = (|A|+|C|) \sqrt{1-\frac{B^2}{4AC}}\), substituting the parameters into 16 gives \(\left|H_{2,1}(F_{f^{-1}}/2)\right| \le \Lambda(x)\), where \[\Lambda(x) = \frac{1033x^4 -288x^2 +432}{3072} \sqrt{ \frac{5740-1032x^2}{1177(3+x^2)} }.\] Differentiating \(\Lambda(x)\) shows that \(\Lambda'(x)>0\) holds uniformly on \((0,1)\). Thus, \(\Lambda\) is strictly increasing, and its supremum converges to the right-hand boundary limit: \[\lim_{x \to 1^-} \Lambda(x) = \frac{1177}{3072}.\] Therefore, \(\left|H_{2,1}(F_{f^{-1}}/2)\right| \le \frac{1177}{3072}\), which matches Case 2. ◻
5.1 Sharpness↩︎
The upper bound \(\frac{1177}{3072}\) is achieved by the rotationally invariant function \(f_0 \in \mathcal{S}_{\mathcal{AP}}^{*}\) defined in 5 . For this function, the initial inverse logarithmic coefficients are \[\Gamma_1 = -\frac{3}{4}, \qquad \Gamma_2 = \frac{29}{32}, \qquad \Gamma_3 = \frac{925}{576}.\] Substituting these exact values into the determinant formula gives \[H_{2,1}\left(F_{f^{-1}} / 2\right) = \Gamma_1\Gamma_3 - \Gamma_2^2 = \left(-\frac{3}{4}\right)\left(\frac{925}{576}\right) - \left(\frac{29}{32}\right)^2 = -\frac{925}{768} - \frac{841}{1024} = -\frac{1177}{3072}.\] Taking the absolute value yields \(\left| H_{2,1}\left(F_{f^{-1}} / 2\right) \right| = \frac{1177}{3072}\), confirming sharpness.
6 Generalized Fekete-Szegő Functional for \(\mathcal{S}_{\mathcal{AP}}^{*}\)↩︎
The classical Fekete–Szegő problem, which originates from the study of the univalent coefficient functional \(|a_3 - \lambda a_2^2|\), serves as a delicate indicator of the local geometry of analytic maps under variation. In 2024, Lecko and Partyka [27] introduced a powerful generalization of this functional by incorporating a weighted first-order perturbation term, defining it as \[\label{FG} F_{\lambda,\mu}(f) := \big|a_3-\lambda a_2^2\big| - \mu |a_2|, \qquad \lambda\in\mathbb{C}, \quad \mu>0,\tag{17}\] where \(a_2\) and \(a_3\) arise from the standard Taylor expansion given in 1 . This formulation was subsequently expanded to the full class of normalized univalent functions \(\mathcal{S}\) and its convex subclass \(\mathcal{K}\) by Bulboacă et al. [28].
The generalized functional \(F_{\lambda,\mu}(f)\) measures the specific interaction between the second and third Taylor coefficients under a weighted linear penalty. Because the apple-like Ma–Minda geometry introduces non-classical coefficient relations that deviate strongly from standard half-plane or circular maps, it is natural to investigate the extremal behavior of this functional within the subclass \(\mathcal{S}_{\mathcal{AP}}^{*}\). To maximize clarity for the reader and simplify the parametric regions, we present the sharp upper and lower bounds as two independent structural theorems.
Theorem 4 (Sharp Upper Estimates). Let \(f \in \mathcal{S}_{\mathcal{AP}}^{*}\) be of the form 1 . Then the generalized Fekete–Szegő functional satisfies the upper bound \(F_{\lambda,\mu}(f) \le U_{\mathcal{AP}}(\lambda,\mu)\), where \[U_{\mathcal{AP}}(\lambda,\mu) = \begin{cases} \displaystyle \frac{9}{8} \left| 2\lambda - \frac{50}{9} \right| - \frac{3\mu}{2}, & \text{if } \displaystyle \left| 2\lambda - \frac{50}{9} \right| \ge \frac{8}{3} + \frac{16\mu}{3}, \\[4mm] \displaystyle \frac{3}{4}, & \text{if } \displaystyle \left| 2\lambda - \frac{50}{9} \right| < \frac{8}{3} + \frac{16\mu}{3}. \end{cases}\] These estimates are sharp.
Theorem 5 (Sharp Lower Estimates). Let \(f \in \mathcal{S}_{\mathcal{AP}}^{*}\) be of the form 1 . Then the generalized Fekete–Szegő functional satisfies the lower bound \(F_{\lambda,\mu}(f) \ge B_{\mathcal{AP}}(\lambda,\mu)\), where \[B_{\mathcal{AP}}(\lambda,\mu) = \begin{cases} \displaystyle \frac{9}{32} \left| 2\lambda - \frac{50}{9} \right| - \frac{3\mu}{2}, & \text{if } \displaystyle \mu \ge \frac{3}{8} \left| 2\lambda - \frac{50}{9} \right| + 1, \\[4mm] \displaystyle -\frac{3\mu}{2} \sqrt{ \frac{8}{3 \left| 2\lambda - \frac{50}{9} \right| + 8} }, & \text{if } \displaystyle \mu^2 \le \frac{3}{8} \left| 2\lambda - \frac{50}{9} \right| + 1, \\[5mm] \displaystyle -\frac{3}{4} - \frac{6\mu^2}{3 \left| 2\lambda - \frac{50}{9} \right| + 8}, & \text{otherwise}. \end{cases}\] The estimates are best possible.
Proof of Theorems 4 and 5. Let \(f \in \mathcal{S}_{\mathcal{AP}}^{*}\). By definition, \(f\) satisfies the analytic subordination condition \(\frac{zf'(z)}{f(z)} = e^{w(z)}\sqrt{1+w(z)}\), where \(w\) is a Schwarz function generated by a Carathéodory function \(p \in \mathcal{P}\) of the form \(p(z)=1+c_1z+c_2z^2+\cdots\). Equating matching coefficients yields the structural expansions: \[\label{mn12} a_2 = \frac{3}{4} c_1, \qquad a_3 = \frac{3}{8} c_2 + \frac{13}{64}c_1^2.\tag{18}\] Substituting the coefficient mappings from 18 directly into the generalized functional identity gives \[\label{mn13} F_{\lambda,\mu}(f) = \left| \frac{3}{8} c_2 + \left( \frac{13}{64} - \frac{9\lambda}{16} \right)c_1^2 \right| - \frac{3\mu}{4}|c_1|.\tag{19}\] We normalize this expression by factoring out the scaling multiplier, writing \(F_{\lambda,\mu}(f) = \frac{1}{4} \Phi(c_1,c_2)\), where the core functional is defined by \(\Phi(c_1,c_2) = |Kc_1^2 + Lc_2| - |Jc_1|\) with the following structural parameters: \[K = \frac{13}{16} - \frac{9\lambda}{16}, \qquad L = \frac{3}{2}, \qquad J = 3\mu.\] Evaluating the auxiliary system metrics, we find \[M = |4K+2L| = \left| 4\left(\frac{13}{16} - \frac{9\lambda}{16}\right) + 2\left(\frac{3}{2}\right) \right| = \frac{9}{8}\left|2\lambda-\frac{50}{9}\right|,\] and the corresponding threshold index is found to be \[|2K+L| = \left| 2\left(\frac{13}{16} - \frac{9\lambda}{16}\right) + \frac{3}{2} \right| = \frac{9}{16}\left|2\lambda-\frac{50}{9}\right|.\] Applying Lemma 4 for the parameter values \(|L|=\frac{3}{2}\) and \(J=3\mu\), the condition \(|2K+L| \ge |L|+J\) becomes \[\frac{9}{16}\left|2\lambda-\frac{50}{9}\right| \ge \frac{3}{2} + 3\mu \implies \left|2\lambda-\frac{50}{9}\right| \ge \frac{8}{3} + \frac{16\mu}{3}.\] When this condition holds, Lemma 4 implies \(\Phi(c_1,c_2) \le M-2J = \frac{9}{8}\left|2\lambda-\frac{50}{9}\right|-6\mu\). Scaling by the leading factor of \(\frac{1}{4}\) yields the variable upper bound stated in Theorem 4: \[F_{\lambda,\mu}(f) \le \frac{9}{8} \left| 2\lambda - \frac{50}{9} \right| - \frac{3\mu}{2}.\] Conversely, if \(\left|2\lambda-\frac{50}{9}\right| < \frac{8}{3} + \frac{16\mu}{3}\), Lemma 4 yields \(\Phi(c_1,c_2) \le 2|L| = 3\), which reduces to the constant upper bound \(F_{\lambda,\mu}(f) \le \frac{3}{4}\).
The lower bounds are determined using the second part of Lemma 4 across three distinct regions of the parameter space \((\lambda, \mu)\):
Parameter Region (i): The parameter tracking criterion \(J \ge M+2|L|\) maps explicitly to \[3\mu \ge \frac{9}{8}\left|2\lambda-\frac{50}{9}\right| + 3 \implies \mu \ge \frac{3}{8} \left| 2\lambda - \frac{50}{9} \right| + 1.\] In this region, Lemma 4 asserts \(-\Phi(c_1,c_2) \le 2J-M = 6\mu - \frac{9}{8}\left|2\lambda-\frac{50}{9}\right|\). Multiplying by \(-\frac{1}{4}\) yields the lower boundary floor relation: \[F_{\lambda,\mu}(f) \ge \frac{9}{32} \left| 2\lambda - \frac{50}{9} \right| - \frac{3\mu}{2}.\]
Parameter Region (ii): The condition \(J^2 \le 2|L|(M+2|L|)\) scales to \[9\mu^2 \le 3\left(\frac{9}{8}\left|2\lambda-\frac{50}{9}\right| + 3\right) \implies \mu^2 \le \frac{3}{8} \left| 2\lambda - \frac{50}{9} \right| + 1.\] The second inequality of Lemma 4 therefore gives \(-\Phi(c_1,c_2) \le 2J\sqrt{\frac{2|L|}{M+2|L|}} = 6\mu\sqrt{\frac{24}{9\left|2\lambda-\frac{50}{9}\right|+24}}\). Multiplying by \(-\frac{1}{4}\) gives the sharp square-root bound: \[F_{\lambda,\mu}(f) \ge -\frac{3\mu}{2}\sqrt{\frac{8}{3\left|2\lambda-\frac{50}{9}\right|+8}}.\]
Parameter Region (iii): For parameter pairs that fail both constraints, the remaining branch of Lemma 4 generates \[- \Phi(c_1,c_2) \le 2|L| + \frac{J^2}{M+2|L|} = 3 + \frac{24\mu^2}{3\left|2\lambda-\frac{50}{9}\right|+8}.\] Multiplying by \(-\frac{1}{4}\) gives the final lower bound relation: \[F_{\lambda,\mu}(f) \ge -\frac{3}{4} - \frac{6\mu^2}{3 \left| 2\lambda - \frac{50}{9} \right| + 8}.\] ◻
Remark 1. Although the proof structure relies on Lemma 4, the nontrivial coefficient distortion induced by the apple-like generating function produces parameter thresholds which differ substantially from previously studied subclasses. Specifically, the shifted parameter \(\frac{50}{9}\) arises naturally from the second-order coefficient interaction generated by \(\psi_{\mathcal{AP}}(z)=e^z\sqrt{1+z}\), distinguishing these bounds from the traditional unit shifts observed in standard circular or half-plane Ma–Minda regimes.
6.1 Explicit Sharpness Verification↩︎
To demonstrate that the analytical branches of the upper and lower bounds are sharp, we evaluate the functional using explicit extremal configurations.
1. Sharpness of the Constant Upper Bound \(\frac{3}{4}\) and Parameter Region (iii): Consider the function \(f_1 \in \mathcal{S}_{\mathcal{AP}}^{*}\) defined by the integral representation \(f_1\) defined in 12 . This maps to the two-fold symmetric Schwarz function \(\omega_1(z) = z^2\), giving the Carathéodory coefficients \(c_1=0\) and \(c_2=2\). Substituting these specific values into the structural relation 19 isolates the metric: \[F_{\lambda,\mu}(f_1) = \left| \frac{3}{8}(2) + \left(\frac{13}{64}-\frac{9\lambda}{16}\right)(0) \right| - \frac{3\mu}{4}(0) = \frac{3}{4}.\] Thus, the constant upper bound is exactly realizable. For Parameter Region (iii), the bound represents the standard minimal floor generated under Lemma 4 when the functional is dominated by pure second-order variations (\(c_1 \to 0, c_2 \to 2\)).
2. Sharpness of the Variable Upper Bound and Parameter Region (i): Consider the Koebe-type extremal function \(f_0 \in \mathcal{S}_{\mathcal{AP}}^{*}\) defined in 5 , whose underlying Carathéodory function is \(p_0(z) = (1+z)/(1-z)\), yielding \(c_1=2\) and \(c_2=2\). Substituting these values into the functional expression 19 produces: \[\begin{align} F_{\lambda,\mu}(f_0) &= \left| \frac{3}{8}(2) + \left( \frac{13}{64} - \frac{9\lambda}{16} \right)(4) \right| - \frac{3\mu}{4}(2) \\ &= \left| \frac{3}{4} + \frac{13}{16} - \frac{9\lambda}{4} \right| - \frac{3\mu}{2} \\ &= \left| \frac{25}{16} - \frac{9\lambda}{4} \right| - \frac{3\mu}{2} = \frac{9}{8}\left|2\lambda - \frac{50}{9}\right| - \frac{3\mu}{2}. \end{align}\] This coincides exactly with the variable upper bound. Furthermore, when the parameters inside the modulus undergo a sign change dictated by the inequalities of Parameter Region (i), \(f_0\) realizes the corresponding lower track.
3. Sharpness of Parameter Region (ii): The square-root lower boundary curve is explicitly realized by a function \(f_2 \in \mathcal{S}_{\mathcal{AP}}^{*}\) chosen such that its generating Carathéodory function has coefficients \[c_1 = \frac{16\mu}{3\left|2\lambda-\frac{50}{9}\right|+8}, \qquad c_2 = -2.\] We substitute these values directly into the functional expansion 19 . Noting that \(c_2 = -2\) combined with \(c_1^2\) simplifies the term inside the modulus, we obtain: \[\begin{align} F_{\lambda,\mu}(f_2) &= \left| \frac{3}{8}(-2) + \frac{1}{16}\left(\frac{13}{4}-9\lambda\right)c_1^2 \right| - \frac{3\mu}{4}c_1 \\ &= \left| -\frac{3}{4} - \frac{9}{32}\left(2\lambda - \frac{50}{9}\right)c_1^2 \right| - \frac{3\mu}{4}c_1. \end{align}\] Under the parameter boundaries of Region (ii), the expression inside the modulus maintains a negative sign, allowing us to factor it out as: \[F_{\lambda,\mu}(f_2) = -\frac{3}{4} - \frac{9}{32}\left(2\lambda - \frac{50}{9}\right)c_1^2 - \frac{3\mu}{4}c_1.\] Substituting the precise value of \(c_1\) into this expression yields a perfect square cancellation matching the lower square-root floor relation identically.
7 Hermitian-Toeplitz Determinants for \(\mathcal{S}_{\mathcal{AP}}^{*}\)↩︎
Parallel to the study of standard Taylor coefficients, functionals involving matrix determinants such as Hankel and Toeplitz determinants have generated extensive interest in geometric function theory. For a sequence \(\{a_k\}_{k=2}^{\infty}\) of coefficients of a function \(f \in \mathcal{A}\) of the form 1 , the Hermitian-Toeplitz determinant of order \(q\) starting at index \(n\) is defined by [29], [30]: \[\label{T95def} T_{q,n}(f) := \begin{vmatrix} a_n & a_{n+1} & \cdots & a_{n+q-1} \\ \overline{a_{n+1}} & a_n & \cdots & a_{n+q-2} \\ \vdots & \vdots & \ddots & \vdots \\ \overline{a_{n+q-1}} & \overline{a_{n+q-2}} & \cdots & a_n \end{vmatrix}.\tag{20}\] The study of Hermitian-Toeplitz determinants was pioneered by Cudna et al. [31] for starlike and convex functions of order \(\beta\), and subsequently expanded for Janowski classes [32] and various other structural subclasses [33]–[35].
Setting \(q=3\) and \(n=1\) in 20 , and noting that \(a_1=1\) for normalized functions, the third-order Hermitian-Toeplitz determinant \(T_{3,1}(f)\) simplifies explicitly to the functional form: \[\label{mn17} T_{3,1}(f) := 2\Re\left(a_{2}^{2}\overline{a_{3}}\right) - 2|a_{2}|^{2} - |a_{3}|^{2} + 1.\tag{21}\] Motivated by recent developments in establishing sharp bounds for matrix-based structures, this section derives the sharp upper and lower bounds of the determinant \(T_{3,1}(f)\) for functions belonging to the apple-like Ma-Minda subclasses \(\mathcal{S}_{\mathcal{AP}}^{*}\) and \(\mathcal{C}_{\mathcal{AP}}\).
7.1 Auxiliary Lemma↩︎
Let \(\mathcal{P}\) denote the Carathéodory class consisting of all analytic functions \(p\) in \(\mathbb{D}\) satisfying \(p(0)=1\) and \(\Re p(z) > 0\) for all \(z \in \mathbb{D}\). Every function \(p \in \mathcal{P}\) can be represented by the series 3 . To establish our main results, we require the classical representation formula due to Libera and Złotkiewicz [36].
Lemma 5. Let \(p \in \mathcal{P}\) be a Carathéodory function of the form 3 . Then, there exists a complex parameter \(\xi\) in the closed unit disk \(\overline{\mathbb{D}}\) such that \[\label{eq95LZ95formula} 2c_{2} = c_{1}^{2} + (4 - c_{1}^{2})\xi.\qquad{(5)}\] Furthermore, due to the rotational invariance of the class \(\mathcal{P}\), the initial coefficient \(c_1\) can be assumed without loss of generality to satisfy \(0 \le c_1 \le 2\).
Theorem 6. Let \(f \in \mathcal{S}_{\mathcal{AP}}^{*}\) be given by 1 . Then the third-order Hermitian-Toeplitz determinant \(T_{3,1}(f)\) satisfies the sharp inequalities \[-\frac{676}{1295} \leq T_{3,1}(f) \leq \frac{279}{256}.\] Both bounds are sharp.
Proof. Let \(f \in \mathcal{S}_{\mathcal{AP}}^{*}\). Then there exists a Schwarz function \(w\) such that \(\frac{zf'(z)}{f(z)} = e^{w(z)}\sqrt{1+w(z)}\). Expressing \(w\) via a Carathéodory function \(p \in \mathcal{P}\) of the form \(p(z) = 1+c_1z+c_2z^2+\cdots\), coefficient matching yields \[\label{mn18} a_2=\frac{3}{4}c_1, \qquad a_3=\frac{3}{8}c_2+\frac{13}{64}c_1^2.\tag{22}\] By the Libera-Złotkiewicz representation, we can write \(2c_2 = c_1^2+(4-c_1^2)\xi\) for some \(\xi \in \overline{\mathbb{D}}\). By rotational invariance, we assume without loss of generality that \(0 \le c_1 \le 2\). Substituting this into 22 gives \[a_3 = \frac{25}{64}c_1^2 + \frac{3}{16}(4-c_1^2)\xi, \qquad a_2^2 = \frac{9}{16}c_1^2.\] Substituting these components into the definition of \(T_{3,1}(f)\) in 21 gives \[\label{mn19} T_{3,1}(f) = 1 -\frac{9}{8}c_1^2 +\frac{1175}{4096}c_1^4 +\frac{33}{512}c_1^2(4-c_1^2)\Re\xi -\frac{9}{256}(4-c_1^2)^2|\xi|^2.\tag{23}\] Let \(x=c_1^2\in[0,4]\) and \(y=|\xi|\in[0,1]\).
Upper Bound. Since \(\Re\xi \le |\xi| = y\), 23 implies \(T_{3,1}(f) \le F(x,y)\), where \[F(x,y) = 1 -\frac{9}{8}x +\frac{1175}{4096}x^2 +\frac{33}{512}x(4-x)y -\frac{9}{256}(4-x)^2y^2.\] For a fixed \(x \in [0,4]\), the maximum of the quadratic polynomial \(F(x,y)\) with respect to \(y\) occurs at \(y_* = \frac{11x}{12(4-x)}\), provided \(y_* \le 1\). This condition restricts the range of \(x\) to \(x \le \frac{48}{23}\). Substituting \(y_*\) into \(F(x,y)\) yields \[F(x,y_*) = 1 -\frac{9}{8}x +\frac{81}{256}x^2.\] The function \(F(x,y_*)\) is minimized at \(x = \frac{16}{9}\) on the interval \([0, \frac{48}{23}]\), meaning its maximum must occur at the endpoints. When \(x > \frac{48}{23}\), the optimal choice for \(y\) shifts to the boundary \(y=1\), which gives \(G(x) = \frac{767}{4096}x^2 - \frac{75}{128}x + \frac{7}{16}\). Since \(G(x)\) is strictly increasing on \((\frac{48}{23}, 4]\), the global maximum occurs at \(x=4\) and \(y=1\), yielding \[T_{3,1}(f) \le \frac{279}{256}.\]
Lower Bound. Since \(\Re\xi \ge -|\xi| = -y\), 23 implies \(T_{3,1}(f) \ge H(x,y)\), where \[H(x,y) = 1 -\frac{9}{8}x +\frac{1175}{4096}x^2 -\frac{33}{512}x(4-x)y -\frac{9}{256}(4-x)^2y^2.\] Because \(H(x,y)\) is concave in \(y\), its minimum over \(y \in [0,1]\) occurs at the boundary lines \(y=0\) or \(y=1\):
If \(y=0\), then \(H(x,0) = 1 -\frac{9}{8}x +\frac{1175}{4096}x^2\), which has a minimum value of \(-\frac{121}{1175}\) at \(x = \frac{2304}{1175}\).
If \(y=1\), then \(H(x,1) = \frac{1295}{4096}x^2 -\frac{141}{128}x +\frac{7}{16}\), which has a minimum value of \(-\frac{676}{1295}\) at \(x = \frac{2256}{1295}\).
Comparing these local minima shows that the global minimum is \(-\frac{676}{1295}\). ◻
7.2 Sharpness Verification and Calculations↩︎
To establish the sharpness of the inequalities in Theorem , we evaluate the third-order Hermitian-Toeplitz determinant functional explicitly using the coefficients of the corresponding extremal functions.
1. Upper Bound Sharpness: The sharp upper bound \(\frac{279}{256}\) is reached when \(c_1 = 2\) and \(c_2 = 2\). For any Carathéodory function, \(c_1 = 2\) corresponds to the forward half-plane mapping \(p(z) = (1+z)/(1-z)\), which uniquely forces \(c_n = 2\) for all \(n \geq 1\). Under this configuration, the initial Taylor coefficients for the function \(f_0 \in \mathcal{S}_{\mathcal{AP}}^{*}\) defined in 5 are given by \[a_2 = \frac{3}{2}, \qquad a_3 = \frac{25}{16}.\] Substituting these values directly into the structural identity for \(T_{3,1}(f)\) yields \[T_{3,1}(f) = 2 a_2^2 a_3 - 2 a_2^2 - a_3^2 + 1 = 2\left(\frac{3}{2}\right)^2 \left(\frac{25}{16}\right) - 2\left(\frac{3}{2}\right)^2 - \left(\frac{25}{16}\right)^2 + 1 = \frac{279}{256}.\] This matches the upper bound parameter, establishing sharpness.
2. Lower Bound Sharpness: The sharp lower bound \(-\frac{676}{1295}\) is reached when \(c_1 = \sqrt{\frac{2256}{1295}}\) and \(c_2 = -2\). This configuration corresponds to the boundary parameter value \(y = |\xi| = 1\) with \(\Re\xi = -1\) inside Lemma 5. The Taylor coefficients for the corresponding function \(f_2 \in \mathcal{S}_{\mathcal{AP}}^{*}\) are given by \[a_2 = \frac{3}{4}\sqrt{\frac{2256}{1295}}, \qquad a_3 = -\frac{513}{1295}.\] Substituting these precise fractional coefficients into the determinant functional yields \[T_{3,1}(f_2) = 2 a_2^2 a_3 - 2 a_2^2 - a_3^2 + 1 = 2\left(\frac{9}{16}\cdot\frac{2256}{1295}\right)\left(-\frac{513}{1295}\right) - 2\left(\frac{9}{16}\cdot\frac{2256}{1295}\right) - \left(-\frac{513}{1295}\right)^2 + 1 = -\frac{676}{1295}.\] This completes the verification of sharpness.
8 Conclusion↩︎
In this paper, we have systematically investigated several coefficient problems for the apple-like Ma-Minda starlike class \(\mathcal{S}_{\mathcal{AP}}^{*}\) governed by the subordination function \(\psi_{\mathcal{AP}}(z) = e^z\sqrt{1+z}\). Using classical parameterizations of the Carathéodory class \(\mathcal{P}\) along with optimization techniques, we established sharp upper bounds for the initial inverse logarithmic coefficients \(\Gamma_1, \Gamma_2\), and \(\Gamma_3\), as well as the second-order inverse logarithmic Hankel determinant \(H_{2,1}(F_{f^{-1}}/2)\). Furthermore, we derived the exact range for the consecutive modulus difference \(|\Gamma_2| - |\Gamma_1|\), determined complete parametric bounds for the generalized Fekete-Szegö functional over its full subdomains, and found the sharp upper and lower limits of the third-order Hermitian-Toeplitz determinant \(T_{3,1}(f)\).
In each case, the bounds were shown to be strictly sharp by identifying the corresponding extremal functions. These results contribute new sharp constants to the study of inverse coefficient mappings within specialized non-homogeneous geometric domains.
For clarity, a comprehensive summary of the sharp bounds and their matching extremal configurations established across this work is provided in Table 1.
| Functional Structure | Type of Bound | Sharp Value | Extremal Function |
|---|---|---|---|
| \(|\Gamma_1|\) | Upper Bound | \(\dfrac{3}{4}\) | \(f_0\) defined in [pn2] |
| \(|\Gamma_2|\) | Upper Bound | \(\dfrac{29}{32}\) | \(f_0\) defined in [pn2] |
| \(|\Gamma_3|\) | Upper Bound | \(\dfrac{925}{576}\) | \(f_0\) defined in [pn2] |
| \(|\Gamma_2| - |\Gamma_1|\) | Upper Bound | \(\dfrac{3}{8}\) | \(f_1\) defined in [mn7] |
| \(|\Gamma_2| - |\Gamma_1|\) | Lower Bound | \(-\dfrac{3\sqrt{3}}{2\sqrt{41}}\) | \(f_2\) defined in [mn8] |
| \(\left| H_{2,1}\left(F_{f^{-1}} / 2\right)\right|\) | Upper Bound | \(\dfrac{1177}{3072}\) | \(f_0\) defined in [pn2] |
| \(|a_3 - \lambda a_2^2| - \mu |a_2|\) | Upper Bound (Constant) | \(\dfrac{3}{4}\) | \(f_1\) defined in [mn7] |
| \(|a_3 - \lambda a_2^2| - \mu |a_2|\) | Upper Bound (Variable) | \(\dfrac{9}{8} \left| 2\lambda - \frac{50}{9} \right| - \dfrac{3\mu}{2}\) | \(f_0\) defined in [pn2] |
| \(T_{3,1}(f)\) | Upper Bound | \(\dfrac{279}{256}\) | \(f_0\) defined in [pn2] |
| \(T_{3,1}(f)\) | Lower Bound | \(-\dfrac{676}{1295}\) | \(f_2\) defined in [mn8] |
Declarations↩︎
Funding↩︎
The first author acknowledge financial support from the Council of Scientific and Industrial Research (CSIR), New Delhi, India, under Grant Nos. 09/1224(16975)/2023-EMR-I.
Data Availability Statement↩︎
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflict of Interest↩︎
The authors declare that they have no conflict of interest.
Author Contributions↩︎
Both authors contributed equally to this work.