On some results of A. E. Livingston and coefficient problems for concave univalent functions (Q616146)

Let \(S(p)\), \(p\in(0,1)\), be the class of univalent functions \(f\), \(f(0)=0\), \(f'(0)=1\), that are meromorphic in the unit disk \(\mathbb D\) and have simple poles at \(z=p\). The family \(\text{Co}(p)\subset S(p)\) consists of the functions \(f\) such that \(\overline{\mathbb C}\setminus f(\mathbb D)\) is convex. For \(f\in\text{Co}(p)\), let \[ f(z)=\sum_{n=-1}^{\infty}a_n(f)(z-p)^n,\;\;\;|z-p|<1-p, \] be the Laurent expansion of \(f\) at \(z=p\), and \[ f(z)=z+\sum_{n=2}^{\infty}b_n(f)z^n,\;\;\;|z|<p, \] be the Taylor expansion of \(f\) at \(z=0\). The author presents a sharp estimate for \(a_1\) in the class \(\text{Co}(p)\) by a method that is different from the method used earlier by Livingston for the same problem. The next theorem proved in the paper gives a sharp inequality involving \(b_2\) and \(b_3\) in the class \(\text{Co}(p)\). Finally, the author states a theorem which describes the domain of variability of \(\mu b_2-b_3\), \(\mu\in\mathbb C\), for \(f\in\text{Co}(p)\). The case \(\mu=1\) yields sharp estimates for \(|b_2-b_3|\) and \(\big||b_2|-|b_3|\big|\) in \(\text{Co}(p)\).