Abstract
In this paper we use a result of Nunokawa to extend some results on univalent functions given by Miller and Mocanu. As a consequence, we get several sufficient conditions for starlikeness over the expression $f(z)f''(z)/f^{\prime2}(z)$ .
Highlights
1 Introduction and preliminaries Let A denote the class of functions f (z) that are analytic in the open unit disk D = {z : |z| < } and are normalized such that f ( ) = f ( ) – =, i.e., f (z) = z + a z + · · ·
For α =, we obtain the well-known class of starlike functions f (z) that map the unit disk onto a starlike region, i.e., if ω ∈ f (D), tω ∈ f (D) for all t ∈ [, ]
A major contribution in the theory of univalent functions was done by the work of Miller and Mocanu
Summary
) Let B(z) and C(z) be complex-valued functions defined in D with ) Let B(z) be a complex-valued function defined in D with arg B(z) ≤ π (z ∈ D). We will extend the result of Theorem . For values of β bigger than β , and we will obtain a result complementary to Theorem .
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