Logarithmic coefficient bounds and coefficient conjectures for classes associated with convex functions (Q2050393)

Summary: It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If \(\mathscr{S}\) denotes the class of functions \(f(z)=z+\sum_{n=2}^\infty a_n z^n\) analytic and univalent in the open unit disk \(\mathbb{U} \), then the logarithmic coefficients \(\gamma_n(f)\) of the function \(f\in\mathscr{S}\) are defined by \(\log(f(z)/z)=2\sum_{n=1}^\infty\gamma_n(f) z^n\). In the current paper, the bounds for the logarithmic coefficients \(\gamma_n\) for some well-known classes like \(\mathscr{C}(1+\alpha z)\) for \(\alpha\in(0,1]\) and \(\mathscr{C} \mathscr{V}_{\mathrm{hpl}}(1/2)\) 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 \(f\in\mathscr{C}(1+\alpha z)\), then the logarithmic coefficients of \(f\) satisfy the inequalities \(|\gamma_n|\leq\alpha/(2n( n + 1)),n\in\mathbb{N}\). Equality is attained for the function \(L_{\alpha,n} \), that is, \(\log( L_{\alpha,n}(z)/z)=2\sum_{n=1}^\infty \gamma_n( L_{\alpha,n}) z^n=(\alpha /n( n + 1)) z^n+\dots,z\in\mathbb{U}\).