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Differential subordination and starlike functions. (English) Zbl 0724.30020
Let \({\mathcal A}\) denote the class of functions f which are analytic in the unit disc \(\Delta\) and normalized in such a way that \(f(0)=0\) and \(f'(0)=1\). For \(0\leq \lambda <1\), denote by \(S^*(\lambda)\), the well- known class of starlike functions of order \(\lambda\). For \(\alpha >0\) and \(\beta <1\), let \[ R(\alpha,\beta)=\{f\in {\mathcal A}:\;Re(f'(z)+\alpha z''(z))>\beta,\quad z\in \Delta \}. \] In this paper among other results we have proved the following: if \[ \delta (\alpha)=\int^{1}_{0}\frac{dt}{1+t^{\alpha}} \] and \(\beta '(\alpha,\alpha ',\beta)=[(\beta -(1-\alpha '/\alpha)(2\delta (\alpha)- 1))]/[(1-(1-\alpha '/\alpha)(2\delta (\alpha)-1))]\), then
(i) \(R(\alpha,\beta '(\alpha,\alpha ',\beta))\subset R(\alpha ',\beta)\), for \(\alpha \geq \alpha '>0\) and \(\beta <1;\)
(ii) \(R(,-\rho)\subset S^*\), for \(\rho\approx -0.0903...;\)
(iii) \(R(,0)\subset S^*(\lambda)\), where \(\lambda\) is the smallest positive root of the cubic equation \[ 4x^ 3-(8 \cos^ 2\theta +3)x+4 \cos^ 2\theta -1=0;\text{ with } \theta =0.911621907; \]
(iv) \(R(\alpha ',0)\subset S^*\), for \(\alpha '\approx 0.42699... \).

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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