Coefficient Problems and Sharp Determinant Estimates for the Class $\mathcal{P}^{*}$

Abstract

We study coefficient problems for the class $\mathcal{P}^{*}$ of normalized analytic functions $f$ in the unit disk satisfying the subordination condition \[ f'(z) \prec e^z(1+\sin z), \qquad z\in\mathbb{D}. \] For functions in this class, we determine sharp bounds for the logarithmic coefficients $\gamma_n$ for $n=1,2,3,4$ and for the inverse logarithmic coefficients $\Gamma_n$ for $n=1,2,3$. In particular, we prove \[ |\gamma_n| \le \frac{1}{n+1} \quad (n=1,2,3,4), \qquad |\Gamma_1|,|\Gamma_2| \le \frac12,\quad |\Gamma_3|\le \frac{35}{48}, \] with equality cases explicitly identified. Moreover, we establish sharp estimates for the moduli of the differences $|\gamma_2|-|\gamma_1|$ and $|\Gamma_2|-|\Gamma_1|$, yielding \[ -\frac12 \le |\gamma_2|-|\gamma_1| \le \frac13, \qquad -\frac{1}{\sqrt{10}} \le |\Gamma_2|-|\Gamma_1| \le \frac13. \] Turning to determinant problems, we obtain sharp upper bounds for the second-order Hankel determinants associated with both the logarithmic and inverse logarithmic coefficients: \[ \bigl|H_{2,1}(F_f/2)\bigr| \le \frac19,\qquad \bigl|H_{2,1}(F_{f^{-1}}/2)\bigr| \le \frac{11}{96}. \] Finally, we derive the sharp two-sided estimate \[ -\frac{9}{35} \le T_{3,1}(f) \le 1 \] for the third-order Hermitian--Toeplitz determinant. All bounds are sharp, and the extremal functions are given explicitly. Our results extend and refine several known coefficient estimates for related subclasses of starlike and convex functions.