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Lobachevskii Journal of Mathematics

, Volume 30, Issue 1, pp 1–11 | Cite as

Main differential sandwich theorem with some applications

Article

Abstract

Let q 1, q 2 be univalent in Δ:= {z: |z| < 1} and p be 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 for 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) ≺ p(z) ≺ q 2(z), where k ∈ ℤ and β ≠ 0, α j ∈ ℂ. Some of its special cases and its applications will be considered for certain analytic functions and certain linear operators.

Key words and phrases

Differential subordinations differential superordinations subordinant dominant 

1991 Mathematics Subject Classification

30C45 

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Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.School of Mathematical Sciences Faculty and Technology University KebangsaanMalaysia BangiMalaysia

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