Abstract
In this paper, we give simple proofs for the bounds (some of them
sharp) of the difference of the moduli of the second and the first logarithmic
coefficient for the general class of univalent functions and for the class of convex
univalent functions.
Results & Lemmas (3)
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Theorem 1
Theorem 1. For every function, holds sharply.
Theorem 1. For every function $f \in \mathcal{S}$ , $-\frac{\sqrt{2}}{2} \leq |\gamma_2| - |\gamma_1| \leq \frac{1}{2}$ holds sharply.
Lemma 1 · coeff
Lemma 1. For all functions from K, The inequality is sharp with extremal function with. Note that the range of for the extremal function…
Lemma 1. For all functions $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ from K,
$$\left|a_3 - a_2^2\right| \le \frac{1}{3} \left(1 - |a_2|^2\right).$$
The inequality is sharp with extremal function
$$f_{\lambda}(z) = \int_{0}^{z} \left(\frac{1-t}{1+t}\right)^{\lambda} \frac{1}{1-t^{2}} dt = z + \lambda z^{2} + \frac{1}{3} (2\lambda^{2} + 1) z^{3} + \cdots,$$
with $0 < \lambda < 1$ .
Note that the range of $\lambda$ for the extremal function exploits the fact that all coefficients in the expansion of convex univalent functions have modulus less or equal to 1.
Theorem 2
Theorem 2. For every function,. The second inequality is sharp.
Theorem 2. For every function $f \in \mathcal{K}$ , $-\frac{1}{\sqrt{10}} \leq |\gamma_2| - |\gamma_1| \leq \frac{1}{6}$ . The second inequality is sharp.