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Zbl 0622.30020
Miller, Sanford S.; Mocanu, Petru T.
Marx-Strohhäcker differential subordination systems.
(English)
[J] Proc. Am. Math. Soc. 99, 527-534 (1987). ISSN 0002-9939; ISSN 1088-6826/e

In this interesting paper, the authors examine generalized Marx- Strohhäcker differential subordinations. Let $\Delta =\{z:\vert z\vert <1\}$. 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: \par 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 [{\it Ch. Pommerenke}, Univalent functions (1975; Zbl 0298.30014)]. The authors use Theorem 1 to prove the following: \par Theorem 2. Let q satisfy the conditions of Theorem 1 and let $$k(z)=z \exp \int\sp{z}\sb{0}((g(t)-1)/t)dt.$$ If $f(z)=z+a\sb 2z\sp 2+..$. 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.
[D.J.Hallenbeck]
MSC 2000:
*30C80 Maximum principle, etc. (one complex variable)
30C45 Special classes of univalent and multivalent functions
34M99 Differential equations in the complex domain
34A40 Differential inequalities (ODE)

Keywords: Marx-Strohhäcker differential subordinations; subordination chains

Citations: Zbl 0456.30022; Zbl 0439.30015; Zbl 0298.30014

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