Abstract
Let the functions be analytic and let 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 are obtained so that either or . As applications, sufficient conditions are determined relating the Schwarzian derivative to the starlikeness or convexity of .
Highlights
Let H U be the class of functions analytic in U : {z ∈ C : |z| < 1} and H a, n be the subclass of H U consisting of functions of the form f z a anzn an 1zn 1 · · ·
Sufficient conditions involving the Schwarzian derivatives of functions f ∈ A are obtained so that zf z /f z is superordinated to a given analytic subordinant q in U
Let Ω be a set in C, q ∈ Q and let n be a positive integer
Summary
Let H U be the class of functions analytic in U : {z ∈ C : |z| < 1} and H a, n be the subclass of H U consisting of functions of the form f z a anzn an 1zn 1 · · ·. Sufficient conditions involving the Schwarzian derivatives are obtained for functions f ∈ A to satisfy either zf z zf z q1 z ≺ f z ≺ q2 z or q1 z ≺ 1 f z ≺ q2 z , 1.10 where the functions q1 are analytic and q2 is analytic univalent in U. Sufficient conditions on functions f ∈ A are obtained so that zf z /f z is subordinated to a given analytic univalent function q in U. We obtained the result 1.7 of Miller and Mocanu 2 relating the Schwarzian derivatives to the starlikeness of functions f ∈ A. Sufficient conditions involving the Schwarzian derivatives of functions f ∈ A are obtained so that zf z /f z is superordinated to a given analytic subordinant q in U.
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