Comprehensive subclsses of analytic functions and coefficient bounds

In this paper, we introduce two general subclasses of analytic functions by means of the principle of subordination and investigate the coefficient bounds for functions in theese classes. The well-known results are obtained as a corollary of our main results. Especially, we improve the results of Altintas and Kilic.


Definitions and Preliminaries
Let A be the family of functions of the form f (z) = z + ∞ n=2 a n z n (1.1) which are analytic in the open unit disk D = {z : z ∈ C and |z| < 1} .
For analytic functions f and g with f (0) = g (0), f is said to be subordinate to g in D if there exists an analytic function h on D such that h (0) = 0, |h (z)| < 1 and f (z) = g (h (z)) (z ∈ D) .
We denote the subordination by Note that if the function g is univalent in D, then we have f (z) ≺ g (z) (z ∈ D) ⇔ f (0) = g (0) and f (D) ⊂ g (D) .
By means of functions belong to the class N and the principle of subordination, we consider following subclasses of analytic function class A: The classes S * (ϕ) and K (ϕ) are introduced by Ma and Minda [5], and the class C (ϕ, ψ) is introduced by Kim et al. [3]. Since we also have in (1.2) and (1.3), then we get the classes of Janowski starlike functions and Janowski convex functions respectively, introduced by Janowski [2].
in (1.4), then we obtain the class of close-to-convex functions of order α and type β, introduced by Libera [4].
in (1.2)-(1.4), then we get the familiar class S * consists of starlike functions in D, K consists of convex functions in D and C consists of close-to-convex function in D, respectively. Also, from (1.5) and (1.6), we get the class CS of close-to-starlike functions in D introduced by Reade [8], and the class Q of quasi-convex functions in D introduced by Noor and Thomas [7], respectively.
Throughout this paper Now we define new comprehensive subclasses of analytic function class A, as follows: where g ∈ K (ψ) .
Remark 5. If we set δ = 0 and λ = 1 in Definition 1 and Definition 2, then we have the classes Also when δ = 0 and λ = 0, we get the classes Remark 6. If we set δ = 0 and and ψ (z) = 1 + z 1 − z in Definition 1 and Definition 2, then we obtain the classes Q CV (λ, A, B) and Q ST (λ, A, B), respectively, introduced very recently by Altıntaş and Kılıç [1]. These classes consist of functions and respectively.
Altıntaş and Kılıç [1] obtained following coefficient bounds for functions belong to the classes Q CV (λ, A, B) and Q ST (λ, A, B), as follows: In this work, we obtain coefficient bounds for functions in the comprehensive subclasses K λ,δ (ϕ, ψ) and S λ,δ (ϕ, ψ) of analytic functions. Our results improve the results of Altıntaş and Kılıç [1] (Theorem 1 and Theorem 2).

Corollaries and Consequences
Letting δ = 0 and λ = 1 in Theorem 3 and Theorem 4, we obtain the following consequences, respectively.

If we choose
in Corollary 2, then we get following consequence.