GENERALIZED CLOSE-TO-CONVEXITY RELATED WITH BOUNDED BOUNDARY ROTATION

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 + 1 2 ) 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.


Introduction
Let A denote the class of analytic functions defined in the open unit disc E = {z : |z| < 1} and be given by (1.1) f (z) = z + ∞ n=2 a n z n , z ∈ E.
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. For details, see [3].
For f, g ∈ A, we say f is subordinate to g in E, written as f (z) ≺ g(z), if there exists a Schwartz function w(z) such that f (z) = g(w(z)), g(z) = z + ∞ n=2 b n z n .
Furthermore, if the function g is univalent in E, then we have the following equivalence f (z) ≺ g(z) ⇔ f (0) = g(0) and f (E) ⊂ g(E).
Convolution of f and g is defined as (g * f )(z) = z + ∞ n=2 a n b n z n .
The class P α [A, B] of strongly Janowski functions is defined as follows. , we refer to [6].
About the class P α [A, B], we observe the following.
This shows φ α (A, B; z) is univalent in E.
This implies Re , if and only if, Special Cases.
where V m is the well known class of functions of bounded boundary rotation. See, for example, [2,10,12].
⊂ R m and R m is the class of functions with bounded radius rotation, see [9].
Then, with f given by (1.1), A n = a n b n , Now, F ∈ V m (ρ), we can write see [13].
Using a result due to Brannan [2], we can write , s 1 , s 2 ∈ S .
Therefore, from (2.1), (2.2) and Cauchy Theorem with z = re ιθ , we have Applying distortion result for s 2 ∈ S and Holder's inequality in (2.3), we get c n z n , we use Parsval identity to have where we have used coefficient bounds |c n | ≤ m(1 − ρ), for h ∈ P m (ρ). From (2.5) together with subordination for starlike functions, and a result due to Hayman [5] for m ≥ 2(1+ρ) 1−ρ , we have where c 1 (m, ρ) denotes a constant.
Special Cases.
(i) Let g(z) = z 1−z , then A n = a n . Take A = 0, and in this case f ∈ V m . This leads us to a known coefficient result that a n = O(1)n ( m 2 −1) .
. Now, using essentially the same method given in [2], the required result follows.
Then, from a result of Goodman [4] and from (2.7), it follows that For this, we can conclude that is analytic and univalent in E. Using the well known Bieberbach Theorem for the best bound for second coefficient of univalent functions, see [3], we have This gives us This completes the proof.
Special Cases.
The following properties of the class V α,m [A, B; g] can easily be proved with simple computations and well known results and therefore we omit the proof.
; g], i = 1, 2 and β, γ are positively real with β + γ = 1. ( (iv) The set of all points log f (z) for a fixed z ∈ E and f ranging over the class V α,m [A, B; g] is convex. ( Theorem 2.5. Let f 1 , f 2 ∈ V α,m [A, B; g], β, δ, c and ν be positively real, Then, for z = re ιθ , 0 ≤ θ 1 < θ 2 ≤ 2π, zF F = p, we have θ2 θ1 Re p(z) Proof. First we show that there exists a function F ∈ A satisfying (2.8). We assume and choose the branches which equal 1, when z = 0. For Hence N is well defined and analytic.

Now let
where we choose the branch of [N (z)] 1 β which equal 1 when z = 0. Thus F ∈ A and satisfies (2.8). We write From (2.8) and (2.9) with some calculations That is We now apply Theorem 2.2 and obtain the required result.

From (3.2) and (3.3), it follows that
and this proves our result.
We note that, for .
Proof. We can write for F ∈ T α,m [A, B; 0; B], Since h ∈ P α [B], it easily follows that We know proceed to prove (ii).
Let d r denote the radius of the largest schlicht disc centered at the origin contained in the image of |z| < r under F (z). Then there is a point z 0 , |z 0 | = r, such that |F (z 0 )| = d r . The ray from 0 to F (z 0 ) lies entirely in the image and the inverse image of this ray is a curve in |z| < r.
Using (3.11), we have Now put t = r 1 u with r 1 = 1+Br 1−Br . Then dt = r 1 du and (3.15) where Γ and G 12 , respectively denote gamma and Gauss hypergeometric functions. Also, here, b, c are positively real for m ≤ 2 1 + 2(1−α) 1−ρ and are given as Using (3.15), (3.16) and (3.17) in (3.14), we obtain the lower bound of |F (z)|. For the upper bound, we proceed in similar way and have Now similar computations yield the required bound and the proof is complete.
By choosing suitable and permissible values of involved parameters, we obtain several new and also known results.
(ii) Let F * be defined as It can be shown, with some computations, that F * belongs to the linearly invariant family of T α,m [A, B; 0; −1].
Using this concept, together with the same argument of Pommerenke [16], we have |A n | ≤ |A * n |, n ≥ 1 and L r (F ) ≤ L r (F * ), F ∈ T α,m [A, B; 0; −1] when L r (F ) is the length of the image of the circle |z| = r under F , 0 ≤ r < 1. , and it is well known that there exists G i ∈ V m such that G (z) = (G (z)) 1−ρ for z ∈ E.
Special Cases.