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Differential subordinations and inequalities in the complex plane. (English) Zbl 0633.34005
Let f and F be analytic in the unit disc U. The function f is subordinate to F, written $f\prec F$ or f(z)$\prec F(z)$, if F is univalent, $f(0)=F(0)$ and f(U)$\subset F(U)$. The authors deal with second order differential subordinations of the form $(1)\quad \psi (p(z),zp'(z),z\sp 2p''(z);z)\prec h(z),$ where $\psi$ : ${\bbfC}\sp 3\times U\to {\bbfC}$. They generalize their previous results [see, Mich. Math. J. 28, 157-171 (1981; Zbl 0439.30015)] on the case (1). With help from this generalization they prove some new inequalities, for example: Theorem 6. If p is analytic in U with $p(0)=0$, then $\vert zp'(z)\vert +z\sp 2p''(z)/p(z)\vert <1$ implies that $\vert p(z)\vert <1;$ Theorem 7. If p is analytic in U with $p(0)=1$, and if $Re[2p(z)- zp''(z)/p'(z)-1]>0,$ then Re p(z)$>0$.
Reviewer: N.V.Grigorenko

34M99Differential equations in the complex domain
Full Text: DOI
[1] Goodman, A. W.: Univalent functions. 1 (1983)
[2] Hallenbeck, D. J.; Ruscheweyh, S.: Subordination by convex functions. Proc. amer. Math. soc. 52, 191-195 (1975) · Zbl 0311.30010
[3] Miller, S. S.; Mocanu, P. T.: Second order differential inequalities in the complex plane. J. math. Anal. appl. 65, 289-305 (1978) · Zbl 0367.34005
[4] Miller, S. S.; Mocanu, P. T.: Differential subordinations and univalent functions. Michigan math. J. 28, 157-171 (1981) · Zbl 0439.30015
[5] Miller, S. S.; Mocanu, P. T.: Univalent solutions of briot-bouquet differential equations. J. differential equations 56, 297-309 (1985) · Zbl 0507.34009