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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 \(\Delta =\{z:| z| <1\}\). If F and G are analytic in \(\Delta\), then F is subordinate to G, written \(F\prec G\) or \(F(z)\prec G(z)\), if G is univalent, \(F(0)=G(0)\) and \(F(\Delta)\subset G(\Delta)\). They first prove the following:
Theorem 1. Let q be univalent in \(\Delta\), with \(q(0)=1\). Set \[ Q(z)=(zq'(z))/(q(z)),\quad h(z)=q(z)+Q(z) \] and suppose that (i) Q is starlike in \(\Delta\), and (ii) \(Re[(zh'(z))/(Q(z))]>0\), \(z\in \Delta\). If B is an analytic function in \(\Delta\) such that \[ B(z) \prec q(z)+(zq'(z))/(q(z)) = h(z), \] then the analytic solution p of \[ zp'(z)+B(z)p(z) = 1\quad (p(0)=1) \] satisfies \(p(z)\prec (1/q(z))\). 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 \[ k(z)=z \exp \int^{z}_{0}((g(t)-1)/t)dt. \] If \(f(z)=z+a_ 2z^ 2+..\). is analytic in \(\Delta\), and \[ (zf''(z))/(f'(z)) \prec (zk''(z))/(k'(z)) \] then \((zf'(z))/(f(z))\) is analytic in \(\Delta\) and \[ (zf'(z))/(f(z)) \prec (zk'(z))/(k(z)). \] 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

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
34M99 Ordinary differential equations in the complex domain
34A40 Differential inequalities involving functions of a single real variable
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