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Main differential sandwich theorem with some applications. (English) Zbl 1169.30302
Summary: Let \(q _{1}, q _{2}\) be univalent in \(\Delta :=\{z: |z|<1\}\), and let \(p\) be a certain analytic function. We give some applications of first order differential subordinations and superordinations to obtain sufficient conditions to satisfy the following sandwich implication which is a generalization of various known sandwich theorems:
\[ \beta zq_1^k (z)q'_1 (z) +\sum\limits_{j = 0}^n {\alpha _j q_1^j (z)} \prec \beta zp^k (z)p'(z)+\sum\limits_{j = 0}^n {\alpha _j p^j (z)} \prec \beta zq_2^k (z)q'_2 (z)+\sum\limits_{j = 0}^n {\alpha _j q_2^j (z)} \]
implies \(q _{1}(z) \prec p(z) \prec q _{2}(z)\), where \(k\in\mathbb Z\), \(\beta\neq 0\), and \(\alpha_j\in\mathbb C\). Some of its special cases and applications are considered for certain analytic functions and certain linear operators.

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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