Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions

Abstract

It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If $\mathcal{S}$ denotes the class of functions $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$ analytic and univalent in the open unit disk $\mathbb{U}$ , then the logarithmic coefficients ${\gamma }_n\left(f\right)$ of the function $f\in \mathcal{S}$ are defined by $\log \left(f\left(z\right)/z\right)=2{\sum }_{n=1}^{\infty }{\gamma }_n\left(f\right)z^n$ . In the current paper, the bounds for the logarithmic coefficients ${\gamma }_n$ for some well-known classes like $\mathcal{C}\left(1+\alpha z\right)$ for $\alpha \in \left(0,1\right]$ and $\mathcal{C}{\mathcal{V}}_{\text{hpl}}\left(1/2\right)$ were estimated. Further, conjectures for the logarithmic coefficients ${\gamma }_n$ for functions $f$ belonging to these classes are stated. For example, it is forecasted that if the function $f\in \mathcal{C}\left(1+\alpha z\right)$ , then the logarithmic coefficients of $f$ satisfy the inequalities $\left|{\gamma }_n\right|\le \alpha /\left(2n\left(n+1\right)\right),n\in \mathbb{N}.$ Equality is attained for the function $L_{\alpha ,n}$ , that is, $\log \left(L_{\alpha ,n}\left(z\right)/z\right)=2{\sum }_{n=1}^{\infty }{\gamma }_n\left(L_{\alpha ,n}\right)z^n=\left(\alpha /n\left(n+1\right)\right)z^n+\cdots ,z\in \mathbb{U}.$