Abstract
The class Pα,m[A, B] consists of functions p, analytic in the open unit disc E with p(0) = 1 and satisfy p(z) = (m/4 + ½) p1(z) – (m/4 – 1/2) p2(z), m ≥ 2, and p1, p2 are subordinate to strongly Janowski function (1+Az/1+Bz)α, α ∈ (0, 1] and −1 ≤ B < A ≤ 1. The class Pα,m[A, B] is used to define Vα,m[A, B] and Tα,m[A, B; 0; B1], B1 ∈ [−1, 0). These classes generalize the concept of bounded boundary rotation and strongly close-to-convexity, respectively. In this paper, we study coefficient bounds, radius problem and several other interesting properties of these functions. Special cases and consequences of main results are also deduced.
Highlights
Let A denote the class of analytic functions defined in the open unit disc
Let S ⊂ A be the class of univalent functions in E and let C, S and K be the subclasses of S consisting of convex, starlike and close-to-convex functions, respectively
The class Pα[A, B] of strongly Janowski functions is defined as follows
Summary
If the function g is univalent in E, we have the following equivalence f (z) ≺ g(z) ⇔ f (0) = g(0) and f (E) ⊂ g(E). The class Pα[A, B] of strongly Janowski functions is defined as follows. An analytic function p : p(z) = 1 + cnzn is in the class Pα,m[A, B], if and only if, there n=1 exist p1, p2 ∈ Pα[A, B] such that (1.2). It is obvious Pα,2[A, B] = Pα[A, B]. Simple calculation yield that (1 − |A|)α−1 Reφα(A, B; z) ≥ α|A − B| (1 − |B|)α+1 > 0, z ∈ E This shows φα(A, B; z) is univalent in E
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