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Zbl 0633.34005
Miller, Sanford S.; Mocanu, Petru T.
Differential subordinations and inequalities in the complex plane. (English)
[J] J. Differ. Equations 67, 199-211 (1987). ISSN 0022-0396

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: \par 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 $\vert p(z)\vert <1;$ \par 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)$>0$.
[N.V.Grigorenko]

MSC 2000:: *34M99 Differential equations in the complex domain

Keywords: second order differential subordinations

Citations: Zbl 0439.30015

Cited in: Zbl 1240.30056 Zbl 1156.33302 Zbl 1090.30016 Zbl 1028.30011 Zbl 0880.30013 Zbl 0867.30006 Zbl 0843.30024 Zbl 0734.30017

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