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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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##### References:
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