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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 $$\alpha '\approx 0.42699...$$.

##### MSC:
 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.)
##### Keywords:
subordination; starlike functions
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