On the difference of the two initial logarithmic coefficients for Bazilevic class of univalent functions
M. Obradovic, N. Tuneski
TL;DR
This work investigates the difference between the moduli of the second and first logarithmic coefficients, $|\gamma_2|$ and $|\gamma_1|$, for Bazilevič type univalent functions in the class $\mathcal{B}_1(\alpha)$, defined by $\Re\left[(f(z)/z)^{\alpha-1} f'(z)\right] > 0$ with $\alpha>0$. It employs a Schwarz-function framework to relate $\gamma_1, \gamma_2$ to the Taylor coefficients $a_2, a_3$ via $\gamma_1 = a_2/2$ and $\gamma_2 = (a_3 - a_2^2/2)/2$, and derives sharp bounds: $-\frac{1}{\sqrt{(\alpha+1)^2+1}} \le |\gamma_2| - |\gamma_1| \le \frac{1}{\alpha+2}$ for all $\alpha>0$, with the right-hand bound global and the left-hand bound sharp only on a subrange $\alpha_1 \le \alpha \le \alpha_2$, where $\alpha_1=\frac{1}{2}(\sqrt{6}-2)$ and $\alpha_2=\sqrt{2}-1$. The proof combines coefficient estimates from the Schwarz function $G(z)$ and explicit extremal constructions to verify sharpness, including a critical threshold $\alpha_3 \approx 0.216$ for the left bound. These results extend prior knowledge on logarithmic coefficients within the univalent function framework and clarify how Bazilevič parameters control extremal behavior.
Abstract
In this paper we give sharp bounds of the difference of the moduli of the second and the first logarithmic coefficient for Bazilevič class of univalent functions.