Some Reciprocal Classes of Close-to-Convex and Quasi-Convex Analytic Functions

The present paper comprises the study of certain functions which are analytic and defined in terms of reciprocal function. The reciprocal classes of close-to-convex functions and quasi-convex functions are defined and studied. Various interesting properties, such as sufficiency criteria, coefficient estimates, distortion results, and a few others, are investigated for these newly defined sub-classes.


Introduction
We denote by A the class of analytic functions on the unit disc U = {z ∈ C : |z| < 1} having the following taylor series representation: The analytic function f will be subordinate to an analytic function g, if there exists an analytic function w, known as a Schwarz function, with w (0) = 0 and |w(z)| < |z|, such that f (z) = g(w(z)).Moreover, if the function g is univalent in U, then we have the following (see [1,2]): f (z) ≺ g(z), z ∈ U ⇐⇒ f (0) = g(0) and f (U) ⊂ g(U).
The classes M (γ) of starlike functions and N (γ) of reciprocal order convex functions γ, (γ > 1) are defined as follows: Using the same concept, together with the idea of k-uniformly starlike and γ ordered convex functions, Nishiwaki and Owa [7] defined the reciprocal classes of uniformly starlike MD (k, γ) and convex functions N D (k, γ).The class MD (k, γ) denotes the subclass of A consisting of functions f satisfying the inequality for some γ (γ > 1) and k (k ≤ 0) and the class N D (k, γ) denotes the subclass of A consisting of functions f (z) satisfying the inequality for some γ (γ > 1) and k (k ≤ 0).They also proved that the well-known Alexander relation holds between MD (k, γ) and N D (k, γ) .This means that For a more detailed and recent study on uniformly convex and starlike functions, we refer the reader to [8][9][10][11][12].
Considering the above defined classes, we introduce the following classes.
Definition 1.Let f belong to A. Then, it will belong to the class KD (β, γ) if there exists g ∈ MD (γ) such that for some β, γ > 1.
Definition 2. Let f belong to A. Then, it will belong to the class QD (β, γ) if there exists g ∈ N D (γ) such that for some β, γ > 1.
It is clear, from (2) and (3), that Definition 3. Let f belong to A. Then, it will belong to the class KD (k, β, γ) if there exists g ∈ MD (k, γ) such that for some k ≤ 0 and β, γ > 1.
Definition 4. Let f belong to A. Then, it is said to be in the class QD (k, β, γ) if there exists g ∈ N D (k, γ) such that for some k ≤ 0 and β, γ > 1.
We can see, from (4) and (5), that the well-known relation of Alexander type holds between the classes KD (k, β, γ) and QD (k, β, γ), which means that

Preliminary Lemmas
Lemma 1.For positive integers t and σ, we have where (σ) t is the Pochhammer symbol, defined by Proof.Using the definition , we write After simplification, we obtain With this, we obtain the required result. where Proof.Let us define a function where p ∈ P, the class of Caratheodory functions (see [1]).One may write Let us write p(z) as p n z n and let f have the series form, as in (1).Then, (11) can be written as After comparing the n th term's coefficients, appearing on both sides, combined with the fact that a 0 = 0, we obtain Now, we take the absolute value and then apply the triangle inequality to get Applying the coefficient estimates, such that |p n | ≤ 2 (n ≥ 1) for Caratheodory functions [1], we obtain where We prove ( 7) by induction on n.Thus, first for n = 2, we obtain the following from ( 12): This proves that, for n = 2, (7) is true.For n = 3, we obtain This proves that when n = 3, (7) holds true.Now, we assume that for t ≤ n, (7) is true, that means Using ( 12) and ( 14), we have After applying (6), we obtain As a result of mathematical induction, it is shown that ( 7) is true for all n ≥ 2. Hence, the required bound is obtained.

Lemma 4 ([13]
).Let w be analytic in U with w(0) = 0.If there exists z 0 ∈ U such that where c is real and c ≥ 1.

Main Results
Theorem 1.
and so we obtain After simplification, we obtain This completes the proof.
In a similar way, one can easily prove the following important result.
where δ k,γ is given by (8) and Proof.If f is in the class KD(k, β, γ), then there exists g(z) ∈ MD (k, γ) such that the function belongs to P. Therefore, we write Let us write p(z) as p(z) = 1 + ∑ ∞ n=1 p n z n , g(z) as g(z) = z + ∑ ∞ n=2 b n z n , and let f (z) have the series form as in (1).Then, (18) can be written as Comparing the nth term's coefficients on both sides, we obtain By taking the absolute value, we get Applying the triangle inequality, we obtain As Re {p(z)} > 0 in U, we have |p n | ≤ 2 (n ≥ 1) (see [1]).Then, from (19), we have where b 1 = 1.Using Lemma (3), we obtain where δ k,β = 2 (β − 1) 1 − k and δ k,γ is defined by (8).This can be written as This completes the proof.
From Definition 4 and Theorem 2, we immediately get the following corollary.
By taking k = 0 in the above results, we obtain the coefficient inequality for the classes KD(β, γ) and QD(β, γ). where Proof.Let f (z) ∈ KD(k, β, γ).Then, there exists g(z) in MD(k, γ) and a Schwarz function w(z) such that as w(z) is analytic U with w(0) = 0 and So, from (22), we obtain This implies that z f (z) which is as required in (20).
Proof.Using Theorem 4, we define the function φ as follows Letting z = re iθ (0 ≤ r < 1), we observe that Let us define As After simplification, we have With the fact that z f (z) g(z) ≺ φ(z), (z ∈ U) and as φ is univalent in U, by using ( 22), we get the required result.
Proof.Let us define a function w(z) by Because φ(z) = 0, we use logarithmic differentiation to get Then, we note that w is analytic in open unit disk and w(0) = 0. Therefore, from (28), we obtain Suppose there exists a point z 0 ∈ U such that max then, by Lemma 4, we can write w(z 0 ) = e iθ and z 0 w (z 0 ) = ce iθ for a point z 0 , and we have which gives that Re z 0 g (z 0 ) g(z 0 ) . By (30), it follows that f ∈ KD(k, β, γ).