## 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

### Keywords:

second order differential subordinations

Zbl 0439.30015
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.