Coefficient problems of Starlike Functions Related to a Balloon-Shaped Domain

Abstract: Recent advances in image and signal processing have drawn on geometric function theory, particularly coefficient estimate problems. Motivated by their significance, we introduce a class of starlike functions related to a balloon-shaped domain [ \mathcal{S}^*_{\mathcal{B}}= \left{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} := B(z); \; z \in \mathbb{D} \right}, ] where $B(z)$ maps the unit disk $\mathbb{D}$ onto a balloon-shaped domain. This work establishes bounds for the second order Hankel determinants and second order Toeplitz determinants involving the initial coefficients, the logarithmic coefficients and the logarithmic coefficients of the inverse function for $f \in \mathcal{S}<sup>*_{\mathcal{B}}$

• The paper introduces a Ma–Minda starlike-function class generated by the balloon-shaped symbol B(z)=1/[1−log(1+z)] and verifies the symbol’s univalence and geometry.
• The paper derives sharp bounds including |a₂|≤1, |a₃|≤3/4, |a₄|≤19/36, |a₅|≤101/288, and a sharp Fekete–Szegö estimate, using Carathéodory and Schwarz-function parametrizations.
• The paper establishes sharp second-order Hankel and Toeplitz determinant bounds for ordinary, logarithmic, and inverse-logarithmic coefficients, with explicit extremal functions attaining the stated constants.

Overview

The paper introduces a new subclass of Ma-Minda starlike functions, denoted $\mathcal{S}^*_{\mathcal{B}}$, defined by the subordination condition

$$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$

where the symbol $B$ maps the unit disk onto a domain whose geometry motivates the name "balloon-shaped" (2602.16208). The domain $B(\mathbb{D})$ is characterized as $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$, with a rightmost tip at $w(0)=1/(1-\log 2)\approx 3.2589$, symmetry about the real axis, and a cusp at the origin. The class fits into the well-developed program initiated by Ma and Minda, in which a univalent symbol $\varphi$ with $\Re\varphi>0$, $\varphi(0)=1$ generates a starlike subclass; the paper situates $\mathcal{S}^*_{\mathcal{B}}$ alongside Janowski, exponential, and other recently introduced geometrically motivated symbols.

The principal contribution is a set of sharp coefficient inequalities for functions in this class: initial Taylor coefficients, the Fekete-Szegö functional, second-order Hankel determinants built from both ordinary and logarithmic coefficients, the analogous determinants for the logarithmic coefficients of the inverse function, and second-order Toeplitz determinants of the same types. All stated bounds are claimed sharp, with extremal functions constructed explicitly via the integral representation

$$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$0

Geometry of the balloon-shaped symbol

The boundary parametrization, obtained by setting $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$1, yields $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$2. The resulting curve is starlike with respect to $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$3, and near the origin its boundary approximates the circle $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$4. This geometry matters technically: the starlikeness and univalence of the symbol are prerequisites for the class to be well-defined within the Ma-Minda framework, and the paper verifies both. Three extremal functions $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$5, $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$6, $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$7 are obtained by substituting $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$8, $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z), \quad z\in\mathbb{D},$$9, and $B$0 into the integral representation; these supply the sharpness witnesses throughout the paper. Notably, $B$1 has only odd powers, which is what makes it the extremal function for the Fekete-Szegö and $B$2 problems, while $B$3 has maximally "rotated" coefficients and serves as extremal for the Toeplitz bounds.

Initial coefficients and the Fekete-Szegö functional

Writing $B$4 for a Schwarz function $B$5, and converting to the Carathéodory parametrization $B$6, the coefficients satisfy

$B$7

From these, the paper derives the sharp bounds

$B$8

all attained by $B$9. The Fekete-Szegö functional satisfies

$B(\mathbb{D})$0

which is sharp for $B(\mathbb{D})$1. Setting $B(\mathbb{D})$2 immediately gives the sharp bound $B(\mathbb{D})$3.

Second-order Hankel determinants

For the ordinary-coefficient determinant, the paper proves

$B(\mathbb{D})$4

with equality for $B(\mathbb{D})$5. The proof exploits rotational invariance of the functional to assume $B(\mathbb{D})$6 (so $B(\mathbb{D})$7), then reduces the estimate to the Choi–Kim–Sugawa functional $B(\mathbb{D})$8 over $B(\mathbb{D})$9, applied with $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$0.

The logarithmic-coefficient Hankel determinant $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$1 is bounded sharply by $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$2, attained by $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$3. This case is the most technically demanding in the paper: here $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$4, so Case 2 of the Choi–Kim–Sugawa lemma applies, and the authors must eliminate several infeasible subcases by verifying sign conditions on explicit polynomials in $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$5. The remaining interval splits at the threshold $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$6; on the second interval the resulting one-variable bound $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$7 peaks at approximately $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$8, below $$\{w\in\mathbb{C}\setminus\{0\}: |\exp(1-1/w)-1|<1\}$$9. The sharpness argument identifies the extremal function via the Carathéodory function $w(0)=1/(1-\log 2)\approx 3.2589$0, recovering $w(0)=1/(1-\log 2)\approx 3.2589$1.

For the inverse function's logarithmic coefficients, the paper establishes

$w(0)=1/(1-\log 2)\approx 3.2589$2

with equality for $w(0)=1/(1-\log 2)\approx 3.2589$3 (corresponding to $w(0)=1/(1-\log 2)\approx 3.2589$4, $w(0)=1/(1-\log 2)\approx 3.2589$5). The proof follows the same subcase-elimination structure, with the split now at $w(0)=1/(1-\log 2)\approx 3.2589$6; the interior bound on the first interval is $w(0)=1/(1-\log 2)\approx 3.2589$7, strictly below the claimed maximum, which is attained at $w(0)=1/(1-\log 2)\approx 3.2589$8. The bound $w(0)=1/(1-\log 2)\approx 3.2589$9 is larger than the direct-function value $\varphi$0, reflecting the additional $\varphi$1 contribution in $\varphi$2.

The logarithmic coefficients themselves satisfy the sharp bounds $\varphi$3, $\varphi$4, $\varphi$5.

Second-order Toeplitz determinants

The Toeplitz results are obtained more directly. For the ordinary coefficients,

$\varphi$6

each sharp, with equality attained by $\varphi$7 in all three cases. The bound for $\varphi$8 requires a genuine two-variable optimization: expressing the coefficients in terms of the Schwarz function's Taylor coefficients $\varphi$9, applying the Zaprawa-type coefficient estimates for Schwarz functions, and maximizing the resulting polynomial $\Re\varphi>0$0 over the region $\Re\varphi>0$1. The paper shows $\Re\varphi>0$2 has no interior critical points, so the maximum lies on the boundary, where the value $\Re\varphi>0$3 is attained, yielding the constant $\Re\varphi>0$4.

For the logarithmic-coefficient Toeplitz determinants, the sharp bounds are

$\Re\varphi>0$5

again both attained by $\Re\varphi>0$6. These follow from one-variable maximization of $\Re\varphi>0$7 and $\Re\varphi>0$8 on $\Re\varphi>0$9 respectively.

Limitations and open questions

The paper concedes, in its motivation, that applications of these determinant estimates to image processing and signal analysis remain limited; the connection to such applications is asserted rather than demonstrated, and no computational experiment is included. Several technical points also merit note. The sharpness of $\varphi(0)=1$0 rests on the boundary case $\varphi(0)=1$1, while the interior estimate is strictly smaller; the proof of the interior bounds relies on the triangle inequality at intermediate steps, so the sharpness of the split-interval analysis depends on the extremal function construction rather than on equality throughout the chain of estimates. The $\varphi(0)=1$2 bound similarly relies on boundary maximization of a polynomial majorant, and the paper does not exhibit an interior equality case beyond the stated extremal function.

The paper leaves open the natural higher-order extensions: third-order Hankel determinants $\varphi(0)=1$3, both for ordinary and logarithmic coefficients, are not treated, nor are the corresponding Toeplitz analogues beyond order two. The radius problems (starlikeness, convexity, and close-to-convexity radii) for $\varphi(0)=1$4 are also not addressed, and no differential-subordination-based refinements of the coefficient bounds are developed.

Conclusion

The paper adds the balloon-shaped domain to the catalog of geometrically motivated Ma-Minda symbols and supplies a complete sharp second-order coefficient analysis for the associated starlike class: initial coefficients, Fekete-Szegö functional, and Hankel and Toeplitz determinants over ordinary, logarithmic, and inverse-logarithmic coefficient families. The methodology—Carathéodory parametrization combined with the Choi–Kim–Sugawa extremal functional and Schwarz coefficient estimates—is standard, but its systematic application to a new symbol, with all bounds verified sharp against explicitly constructed extremal functions, constitutes the paper's substantive contribution.