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LetA,B,D,E∈[−1,1]and letp(z)be an analytic function defined on the open unit disk,p(0)=1. Conditions onA,B,D, andEare determined so that1+βzp'(z)being subordinated to(1+Dz)/(1+Ez)implies thatp(z)is subordinated to(1+Az)/(1+Bz). Similar results are obtained by considering the expressions1+β(zp'(z)/p(z))and1+β(zp'(z)/p2(z)). These results are then applied to obtain sufficient conditions for analytic functions to be Janowski starlike.

Abstract. Differential subordination and superordination results are obtained for analytic functions in the open unit disk which are associated with the multiplier transformation. These results are obtained by investigating appropriate classes of admissible functions. Sandwich-type results are also obtained. (2000): 30C80, 30C45. Mathematics subject classification

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Let the functions q 1 be analytic and let q 2 be analytic univalent in the unit disk. Using the methods of differential subordination and superordination, sufficient conditions involving the Schwarzian derivative of a normalized analytic function f are obtained so that either q 1 z ≺ zf z /f z ≺ q 2 z or q 1 z ≺ 1 zf z /f z ≺ q 2 z . As applications, sufficient conditions are determined relating the Schwarzian derivative to the starlikeness or convexity of f.