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On the order of starlikeness of the class UST. (English) Zbl 0931.30005
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 $$\text{UST}= \{f\in A: \text{Re} [(z-\zeta) f'(z)/ (f(z)- f(\zeta))]\geq 0,\ (z,\zeta)\in D\times D\}.$$ {\it 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\ $.
MSC:
30C45Special classes of univalent and multivalent functions
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References:
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