On strongly starlike and strongly convex functions with bounded radius and bounded boundary rotation

In this paper, we introduce and investigate new classes of normalized analytic functions in an open unit disk with bounded radius and bounded boundary rotation by using the subordination. We discuss inclusion results, co-efficient bounds, growth and distortion theorems of the classes. Moreover, we compute the radii of strong starlikeness, convexity and starlikeness of the classes. It is interesting to mention that most of our findings are best possible as compared to the existing results in the literature.


Introduction and preliminaries
We represent A as the class of functions f (z), which is analytic in E = {z ∈ C : |z| < 1}. Then, for each z ∈ E, f (z) has the form f (z) = z + ∞ n=2 a n z n . (1.1) Suppose that g(z) and G(z) are analytic functions in E. Then g(z) is said to be subordinate to G(z) written as g(z) ≺ G(z) if and only if there exists w(z), which is analytic in E with w(0) = 0 and |w(z)| < 1. Therefore, g(z) = G(w(z)) belongs to E. g(z) is univalent in E follows that g ≺ G is equivalent to g(0) = G(0) and g(E) ⊂ G(E). The function F(A, B, z) = ( 1+Az 1+Bz ) is the conformal mapping of the unit disk to circle, which is symmetric with respect to the real axis having center 1-AB 1-B 2 and radius A-B 1-B 2 for each A, B such that -1 ≤ B < A ≤ 1. A function p(z) (with p(0) = 1) is analytic in E and belongs to the class P [A, B] if p(z) is subordinate to F(A, B, z). In [5], Janowski introduced and investigated the class P [A, B]. Later on, Noor and Arif [8] investigated the class P m [A, B]. They investigated that a function p(z) (with p(0) = 1) is analytic in E and belongs to the class P m [A, B] if and only if there exist p 1 (z), p 2 (z) ∈ P[A, B] such that In [14], Rajapat et al. used the known family of fractional integral operators (with the Gauss hypergeometric function in the kernel) and defined new subclasses of strongly starlike and strongly convex functions of order β and type α in the open unit disk U. Moreover, they established several inclusion relationships and interesting results associated with the fractional integral operators.
Shiraishi et al. [16] investigated some new sufficient conditions for the class of strong Caratheodory functions in the open unit disk U. Example and several corollaries of the main results were presented.
We consider a function , and 0 < β ≤ 1, which is analytic and univalent in E. If p(z) (with p(0) = 1) is subordinate to F β (A, B, z), then the function p(z) is analytic in E and belongs to the classP β [A, B].

Main results
In this section, we discuss the coefficient problem, analytic property, inclusion results, and radius problem. It is interesting to mention that our obtained results are sharp as compared to the existing results in the literature. First we define the following.

Definition 2.2 An analytic function
if and only if f is locally univalent and Note that, for particular cases of A, B, and β, several classes are obtained and investigated in [8]. For β = 1 2 , m = 2, B = 0 and A = a, a ∈ (0, 1], Aouf [1] obtained the following class: Now we prove the following lemmas, which represent the coefficient inequality of functions belonging to the classP Then If q 1 (z) = 1 + ∞ n=1 b n z n and q 2 (z) = 1 + ∞ n=1 c n z n , then 1 + ∞ n=1 a n z n = m By comparing the coefficients of z n and using the triangle inequality, we get Using the well-known result of Rogosinski [15] on subordination, we have |b n | ≤ β|A -B| and |c n | ≤ β|A -B| for all n ≥ 1. This implies that which is as required. This completes the proof.
The following lemma extends the inequality bound of functions in the class P m [A, B].

Lemma 2.5 If q(z) belongs toP
Then Using the triangle inequality, we have where w(z) is defined in E, which is an analytic function with w(0) = 0 and |w(z)| < r. Using (2.8) in (2.7), we get the right-hand side of (2.5). Moreover, (2.6) can be written as and (2.11*) By simple calculations, we get the left-hand side of (2.5). Hence the proof.

Lemma 2.6 If a function q(z) ∈ A belongs toP
Proof Let q(z) belong toP In the following theorems, we discuss and investigate the coefficient problem, analytic property, inclusion results, and radius problem.

Theorem 2.7 If f ∈Ṽ
(2.14) Brannan [2] showed that each g(z) ∈ V m has the representation of the form , s 1 , s 2 ∈ * S. (2.15) It has been proved in [12] that each σ i ∈ * S(γ ) is  Comparing the coefficient of z n , we obtain (n -1)a n = n-1 i=1 c i a n-i , c 0 = 1 and hence For n = j, where we use the Pochhammer notation (β) m introduced in [11]: Consider Using (2.20), we obtain the following: By induction, we get (2.17). Hence the proof.
B] with f (z) = z + ∞ n=2 a n z n , z ∈ E. Then where q(z) is analytic in E with q(0) = 1. From (2.21), we have Using (2.23) and (2.24), we get with η 1 = 1 1-ρ , η 2 = ρ 1-ρ . Using (2.22) with the convolution techniques given by Noor [7], we have Thus, using (2.25) and (2.26), we have Since the first two conditions of Lemma 1.2 are obviously satisfied, we satisfy condition (iii) as follows: It is observed that (2.28) gives negative value for A ≤ 0 and B ≤ 0. Therefore, using A ≤ 0, we have and from B ≤ 0, we obtain 0 ≤ ρ < 1. Hence all three conditions of Lemma 1.2 are satisfied. From this it follows that q i (z) ∈ P for i = 1, 2 and z ∈ E. Therefore q(z) ∈ P m and f ∈ R m (ρ), where ρ is given in (2.29). This completes the proof.

Conclusion
We introduced some new classes of analytic functions with bounded radius and bounded boundary rotation by using the subordination. We discussed inclusion results, coefficient bounds, growth and distortion theorems of the classes. The radii of starlikeness and strong starlikeness of the classes have been computed. It is observed that most of the results are best possible.