Janowski type close-to-convex functions associated with conic regions

The analytic functions, mapping the open unit disk onto petal and oval type regions, introduced by Noor and Malik (Comput. Math. Appl. 62:2209-2217, 2011), are considered to define and study their associated close-to-convex functions. This work includes certain geometric properties like sufficiency criteria, coefficient estimates, arc length, the growth rate of coefficients of Taylor series, integral preserving properties of these functions.


Introduction and definitions
For two functions f and g analytic in E, we say that f is subordinate to g, denoted by f ≺ g, if there exists a Schwarz function w with w() =  and |w(z)| <  such that f (z) = g(w(z)).
In particular, if g is univalent in E, then f () = g() and f (E) ⊂ g (E). For more details, see []. Consider the domain For fixed k, k represents the conic region bounded successively by the imaginary axis (k = ), the right branch of a hyperbola ( < k < ), a parabola (k = ) and an ellipse (k > ). This domain was studied by Kanas [-]. The function p k , with p k () = , p k () >  plays the role of extremal and is given by . Let P p k denote the class of all those functions p(z) which are analytic in E with p() =  and p(z) ≺ p k (z) for z ∈ E. Clearly, it can be seen that P p k ⊂ P, where P is the class of functions with a positive real part (see [, ]). For the applications and exclusive study of the class P, we refer to [-]. More precisely and, for p ∈ P p k , we have Therefore, we can write equivalently, we can write These two classes were recently introduced by Noor and Malik []. Motivated by the recent work presented by Noor and Malik [], we define some classes of analytic functions associated with conic domains as follows.
It can easily be seen that . Throughout this paper, we assume that - ≤ D < C ≤ , - ≤ B < A ≤  and k ≥  unless otherwise specified.

A set of lemmas
To prove our main results, we need the following lemmas. and ) Let f and g be in the class C and S * , respectively. Then, for every function F(z) analytic in E with F() = , we have where " * " denotes the well-known convolution of two analytic functions and coF(E) denotes the closed convex hull F(E).

Lemma . ([]) Let g ∈ k-ST [C, D] with k ≥  and be given by
Then where δ k is defined by (.).

The main results and their consequences
This Hence (.) becomes which implies that This completes the proof.
which has the form (.). Then, for n ≥ , The last inequality is bounded by  if Hence we have The proof follows immediately by using Theorem . and relation (.).

Corollary . ([]) A function is said to be in the class
. Necessary condition where λ is defined by (.) .
We can write Also, we observe that, for h ∈ P[A, B], Using (.) and (.) in (.), we obtain This completes the required result.
where β  is defined by (.) and C (λ, A, B) is a constant depending upon λ, A and B. Proof Let . We can write Using Holder's inequality, we have Using (.) and the distortion result for a starlike function in (.), we obtain where C(λ, A, B) = π λ  (A -B) λ and (β  ) + λ > . This completes the proof. Proof To prove the result, we need to prove