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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 \((1)\quad \psi (p(z),zp'(z),z^ 2p''(z);z)\prec h(z),\) where \(\psi\) : \({\mathbb{C}}^ 3\times U\to {\mathbb{C}}\). 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 \(| zp'(z)| +z^ 2p''(z)/p(z)| <1\) implies that \(| p(z)| <1;\)
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

MSC:

34M99 Ordinary differential equations in the complex domain

Citations:

Zbl 0439.30015
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References:

[1] Goodman, A.W, ()
[2] Hallenbeck, D.J; Ruscheweyh, S, Subordination by convex functions, (), 191-195 · 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
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