On the order of starlikeness of the class UST (Q1301877)

Let \(A\) denote the class of functions \(f(z)= z+ \sum_{k=2}^\infty a_k(f)z^k\), regular and normalized in the unit disk \(D\). The author studies the relation between two subclasses of \(A\), the class of starlike functions of order \(\alpha\), \(\alpha<1\), \[ \text{ST}(\alpha)= \{f\in A: \text{Re} [zf'(z)/ (f(z)]\geq \alpha,\;z\in D\} \] and the class of uniformly starlike functions \[ \text{UST}= \{f\in A: \text{Re} [(z-\zeta) f'(z)/ (f(z)- f(\zeta))]\geq 0,\;(z,\zeta)\in D\times D\}. \] \textit{F. Rønnig} [J. Math. Anal. Appl. 194, No. 1, 319-327 (1995; Zbl 0834.30011)] showed that \(\text{UST}\not\subset \text{ST}(\frac 12)\) and posed the problem of determining the largest \(\alpha\geq 0\) such that \(\text{UST} \subset \text{ST} (\alpha)\). The author proved that if \(\alpha> \alpha_0= 0.1483\dots\), then \(\text{UST} \not\subset \text{ST}(\alpha)\). The bound is determined as \(\alpha_0= 1-h_0^{-1/2}\), where \(h_0= 1.3786\dots\) is the maximum of the function \[ h(s,t)= \tfrac 14 \Bigl[1+ st+ \sqrt{(1-s^2) (1-t^2)} \Bigr] \Bigl[1+ st+ \sqrt{(1-t^2) (1+t^2+ 2st)}\Bigr] \] in the square \(0\leq s,t\leq 1\) which is attained for \(s_0= 0.9246\dots\), \(t_0= 0.7803\dots\;\).