Moduli difference of initial inverse logarithmic coefficients for starlike and convex functions

Abstract

Let $\mathcal{A}$ denote the class of functions $f$ that are analytic in the open unit disk $\mathbb{D}$ and satisfy the normalization conditions $f(0) = 0$ and $f'(0) = 1$. This paper investigates the inverse logarithmic coefficients $Γ_n$, which are defined by the expansion $\log(f^{-1}(w)/w) = 2\sum_{n=1}^{\infty} Γ_n w^n$. We establish sharp upper and lower bounds for the difference of the moduli of the first two inverse logarithmic coefficients, $|Γ_2| - |Γ_1|$, for several significant subclasses of univalent functions. Specifically, we derive sharp estimates for functions belonging to the class of starlike functions with respect to symmetric points ($\mathcal{S}_S^*$), convex functions with respect to symmetric points ($\mathcal{K}_S$), and functions associated with the lune domain ($\mathcal{S}_{\leftmoon}^*$ and $\mathcal{C}_{\leftmoon}$). The results are obtained by employing subordination techniques and utilizing sharp estimates for the coefficients of Schwarz functions. In each case, the extremal functions that attain these bounds are explicitly identified. Our findings provide further insights into the geometric properties of inverse mappings and extend recent research on coefficient functionals in geometric function theory.

Figures (5)

• Figure 1: The figure focuses strictly on the Lune domain $\mathcal{L}$ defined by the inequality $|w^2 - 1| \le 2|w|$. The region is bounded by two symmetric arcs that meet at the vertical vertices $i$ and $-i$. On the real axis, the right-hand lobe extends from $1 - \sqrt{2} \approx -0.414$ to $1 + \sqrt{2} \approx 2.414$. Due to symmetry, the left-hand lobe spans from $-(1 + \sqrt{2})$ to $-(1 - \sqrt{2})$. This region is the range of the function $q(z) = z + \sqrt{1+z^2}$ for $z$ in the unit disk $\mathbb{D}$. In geometric function theory, a function $f$ belongs to the class $\mathcal{S}_{\leftmoon}^{*}$ if the quantity $\frac{zf'(z)}{f(z)}$ takes values within this domain.
• Figure 2: The graph exhibits the image domain of $f_1(\mathbb{D})$.
• Figure 3: The graph exhibits the image domain of $f_2(\mathbb{D})$.
• Figure 4: The graphs exhibit the image domains of $f_3(\mathbb{D})$ and $f_4(\mathbb{D})$.
• Figure 5: The graphs exhibit the image domains of $f_5(\mathbb{D})$ and $f_6(\mathbb{D})$.

Theorems & Definitions (10)

• Definition 1.1
• Theorem 2.1
• Lemma 2.1
• proof : Proof of Theorem \ref{['Th-1.1']}
• Theorem 2.2
• proof : Proof of Theorem \ref{['Th-1.2']}
• Theorem 2.3
• proof : Proof of Theorem \ref{['Th-1.3']}
• Theorem 2.4
• proof : Proof of Theorem \ref{['Th-1.4']}