Lobachevskii Journal of Mathematics

, Volume 30, Issue 1, pp 1–11

# 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

30C45

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