Fekete-Szegő problem for some starlike functions related to shell-like curves (Q2810998)

The paper is related to the geometric theory of analytic functions. Let \(\mathcal {A}\) denote the class of complex functions \(f\) of the form \(f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}\) which are analytic in \(\mathbb {U}:=\{z\in \mathbb {C}\:\left | z\right | <1\}\).NEWLINENEWLINEA typical problem in geometric function theory is to study a functional made up of combinations of the coefficients of the original function. Usually, there is a parameter over which the extremal value of the functional is needed. The paper deals with one important functional of this type: the Fekete-Szegő functional defined by \(\Lambda_{\mu}\left (f\right)=a_{3}-\mu a_{2}^{2},\;\;f\in \mathcal {A}\). The problem of maximizing the absolute value of the functional \(\Lambda_{\mu}\) in subclasses of normalized functions is called the Fekete-Szegő problem.NEWLINENEWLINEIn 1999 the second author [Zesz. Nauk. Politech. Rzesz. 175, Mat. 23, 111--116 (1999; Zbl 0995.30012)] introduced the class \(\mathcal {SL}\) of functions \(f\in \mathcal {A}\) satisfying the condition NEWLINE\[NEWLINE \frac {zf^{\prime}(z)}{f(z)}\prec \frac {1+\tau ^{2}z^{2}}{1+\tau z-\tau ^{2}z^{2}}, NEWLINE\]NEWLINE where ``\(\prec\)'' denotes subordination and the number \(\tau =(\sqrt {5}-1)/2\) fulfils the golden section of the segment \([0,1]\). For \(f\in \mathcal {SL}\) the boundary of \(f(\mathbb {U})\) is related to the conchoid of de Sluze and the coefficients \(a_{n}\) are associated with the Fibonacci numbers.NEWLINENEWLINEThe main result of the paper is the solution of the Fekete-Szegő problem in the class \(\mathcal {SL}\).