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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 ${\Delta }=\left\{z:|z|<1\right\}$. If F and G are analytic in ${\Delta }$, then F is subordinate to G, written $F\prec G$ or $F\left(z\right)\prec G\left(z\right)$, if G is univalent, $F\left(0\right)=G\left(0\right)$ and $F\left({\Delta }\right)\subset G\left({\Delta }\right)$. They first prove the following:

Theorem 1. Let q be univalent in ${\Delta }$, with $q\left(0\right)=1$. Set

$Q\left(z\right)=\left(z{q}^{\text{'}}\left(z\right)\right)/\left(q\left(z\right)\right),\phantom{\rule{1.em}{0ex}}h\left(z\right)=q\left(z\right)+Q\left(z\right)$

and suppose that (i) Q is starlike in ${\Delta }$, and (ii) $Re\left[\left(z{h}^{\text{'}}\left(z\right)\right)/\left(Q\left(z\right)\right)\right]>0$, $z\in {\Delta }$. If B is an analytic function in ${\Delta }$ such that

$B\left(z\right)\prec q\left(z\right)+\left(z{q}^{\text{'}}\left(z\right)\right)/\left(q\left(z\right)\right)=h\left(z\right),$

then the analytic solution p of

$z{p}^{\text{'}}\left(z\right)+B\left(z\right)p\left(z\right)=1\phantom{\rule{1.em}{0ex}}\left(p\left(0\right)=1\right)$

satisfies $p\left(z\right)\prec \left(1/q\left(z\right)\right)$. 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

$k\left(z\right)=zexp{\int }_{0}^{z}\left(\left(g\left(t\right)-1\right)/t\right)dt·$

If $f\left(z\right)=z+{a}_{2}{z}^{2}+··$. is analytic in ${\Delta }$, and

$\left(z{f}^{\text{'}\text{'}}\left(z\right)\right)/\left({f}^{\text{'}}\left(z\right)\right)\prec \left(z{k}^{\text{'}\text{'}}\left(z\right)\right)/\left({k}^{\text{'}}\left(z\right)\right)$

then $\left(z{f}^{\text{'}}\left(z\right)\right)/\left(f\left(z\right)\right)$ is analytic in ${\Delta }$ and

$\left(z{f}^{\text{'}}\left(z\right)\right)/\left(f\left(z\right)\right)\prec \left(z{k}^{\text{'}}\left(z\right)\right)/\left(k\left(z\right)\right)·$

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

MSC:
 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable) 30C45 Special classes of univalent and multivalent functions 34M99 Differential equations in the complex domain 34A40 Differential inequalities (ODE)