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

Abstract

We study coefficient problems for the class ${\mathcal{P}}^{\ast }$ 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 \[ |γ_n| \le \frac{1}{n+1} \quad (n=1,2,3,4), \qquad |Γ_1|,|Γ_2| \le \frac12,\quad |Γ_3|\le \frac{35}{48}, \] 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 \[ -\frac12 \le |γ_2|-|γ_1| \le \frac13, \qquad -\frac{1}{\sqrt{10}} \le |Γ_2|-|Γ_1| \le \frac13. \] 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.

24 pages, 1 figure