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Marx-Strohhäcker differential subordination systems. (English) Zbl 0622.30020

In this interesting paper, the authors examine generalized Marx- Strohhäcker differential subordinations. Let Δ={z:|z|<1}. If F and G are analytic in Δ, then F is subordinate to G, written FG or F(z)G(z), if G is univalent, F(0)=G(0) and F(Δ)G(Δ). They first prove the following:

Theorem 1. Let q be univalent in Δ, with q(0)=1. Set

Q(z)=(zq ' (z))/(q(z)),h(z)=q(z)+Q(z)

and suppose that (i) Q is starlike in Δ, and (ii) Re[(zh ' (z))/(Q(z))]>0, zΔ. If B is an analytic function in Δ such that

B(z)q(z)+(zq ' (z))/(q(z))=h(z),

then the analytic solution p of

zp ' (z)+B(z)p(z)=1(p(0)=1)

satisfies p(z)(1/q(z)). The proof uses a lemma proved by the authors [Mich. Math. J. 28, 151-171 (1981; Zbl 0439.30015)] and a lemma on subordination chains found in [Ch. Pommerenke, Univalent functions (1975; Zbl 0298.30014)]. The authors use Theorem 1 to prove the following:

Theorem 2. Let q satisfy the conditions of Theorem 1 and let

k(z)=zexp 0 z ((g(t)-1)/t)dt·

If f(z)=z+a 2 z 2 +··. is analytic in Δ, and

(zf '' (z))/(f ' (z))(zk '' (z))/(k ' (z))

then (zf ' (z))/(f(z)) is analytic in Δ and

(zf ' (z))/(f(z))(zk ' (z))/(k(z))·

The authors give many examples as applications of these two theorems. In the last part of the paper, they consider differential subordinations with starlike superordinate functions.

Reviewer: D.J.Hallenbeck

30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30C45Special classes of univalent and multivalent functions
34M99Differential equations in the complex domain
34A40Differential inequalities (ODE)