Some subclasses of analytic functions of complex order

Abstract: In this paper, we introduce and investigate two new subclasses of analytic functions in the open unit disk in the complex plane. Several interesting properties of the functions belonging to these classes are examined. Here, sufficient, and necessary and sufficient, conditions for the functions belonging to these classes, respectively, are also given. Furthermore, various properties like order of starlikeness and radius of convexity of the subclasses of these classes and radii of starlikeness and convexity of these subclasses are examined.


Introduction and preliminaries
Let A be the class of analytic functions f (z) in the open unit disk U = {z ∈ C : |z| < 1} , normalized by f (0) = 0 = f ′ (0) − 1 of the form f (z) = z + a 2 z 2 + a 3 z 3 + · · · + a n z n + · · · = z + ∞ ∑ n=2 a n z n , a n ∈ C, (1.1) and S denote the class of all functions in A that are univalent in U . Also, let us define by T the subclass of all functions f (z) in A of the form f (z) = z − a 2 z 2 − a 3 z 3 − · · · − a n z n − · · · = z − ∞ ∑ n=2 a n z n , a n ≥ 0. (1.2) Furthermore, we will denote by S * (α) and C(α) the subclasses of S that are, respectively, starlike and convex functions of order α (α ∈ [0, 1)). By definition (see for details [4,5] and also [9]), and 1) . (1.4) For convenience, S * = S * (0) and C = C(0) are, respectively, starlike and convex functions in U . It is easy to verify that C ⊂ S * ⊂ S . For details on these classes, one could refer to the monograph by Goodman [5].
Inspired by the aforementioned works, we define a subclass of analytic functions as follows.

Coefficient bounds for the classes S * C(α, β; τ ) and T S * C(α, β; τ )
In this section, we will examine some inclusion results of the subclasses S * C(α, β; τ ) and T S * C(α, β; τ ) of analytic functions in the open unit disk. Also, we give coefficient bound estimates for the functions belonging to these classes. A sufficient condition for the functions in class S * C(α, β; τ ) is given by the following theorem.
It suffices to show that 1 τ Considering (1.1), by simple computation, we write

The last expression is bounded above by
Thus, the proof of Theorem 2.1 is completed. 2 Setting τ = 1 in Theorem 2.1, we arrive at the following corollary.
if the following condition is satisfied: The result is sharp for the functions z n , z ∈ U, n = 2, 3, ... . [8].

Remark 2.3 The result obtained in Corollary 2.2 verifies Corollary 2.1 in
Choosing β = 0 in Corollary 2.2, we have the following result.

Corollary 2.4 (see [8, p. 110, Theorem 1]) The function f (z) by definition by (1.1) belongs to the class
The result is sharp for the functions Taking β = 1 in Corollary 2.2, we arrive at the following result.
The result is sharp for the functions [7], respectively.

Remark 2.6 The results obtained in Corollary 2.4 and 2.5 verify Corollary 2.3 and 2.4 in
From the following theorem, we see that for the functions in the class The result is sharp for the functions

Proof
The proof of the sufficiency of the theorem can be proved similarly to the proof of Theorem 2.1. Therefore, we will prove only the necessity of the theorem.

Using (1.2) and (2.3), we can easily show that
Re Thus, from the previous inequality, letting z → 1 through real values, we have We will examine the last inequality depending on the different cases of the sign of parameter τ as follows.
Choosing β = 1 in Corollary 2.8, we arrive at the following result.

Remark 2.12
The results obtained in Corollary 2.10 and 2.11 verify Corollary 2.7 and 2.8 in [7], respectively.
On the coefficient bound estimates of the functions belonging in the class T S * C(α, β; τ ) , we give the following theorem.

Theorem 2.15 Let the function f (z) by definition by (1.2) belong to the class T S
and . (2.8) Proof Using Theorem 2.7, we obtain From here, we can easily show that (2.7) is true.
Similarly, we write that is, Using (2.7) in the last inequality, we obtain which immediately yields inequality (2.8).
Thus, the proof of Theorem 2.15 is completed. 2 Setting τ = 1 in Theorem 2.15, we obtain the following corollary. .
From Theorem 2.7, for the coefficient bound estimates, we arrive at the following result.

Remark 2.19
Numerous consequences of Corollary 2.14 can indeed be deduced by specializing the various parameters involved. Many of these consequences were proved by earlier workers on the subject (see, for example, [1,8,10]).

Order of starlikeness and radius of convexity for the classes T C(α; τ ) and T S * (α; τ )
In this section, we will examine some properties like order of starlikeness and radius of convexity of the subclasses T C(α; τ ) and T S * (α; τ ) . On this, we can give the following theorem.
Then the function f (z) belongs to the class T S * (α 0 ; τ ) , Proof In view of Corollary 2.13 and Corollary 2.14, we must prove that |τ | a n ≤ 1 It suffices to show that for all n = 2, 3, ... .

The last inequality is equivalent to
Taking into account that α ≥ 0 , it suffices to show that which is equivalent to n (n − 1) ≥ 2 (n − 1); that is, n ≥ 2. Thus, inequality (3.2) is provided for all n = 2, 3, ... . With this the proof of Theorem 3.1 is completed.  . The result is sharp for the functions z n , z ∈ U, n = 2, 3, ... ..

Note 3.1. There is no converse to Theorem 3.1. That is, a function in T S * (α; τ ) need not be convex. To
show this, we need only find coefficients a n , n = 2, 3, ... for which + 1 all satisfy both inequalities of ] is the exact value of number x .
By considering the above note, we now determine the radius of convexity for functions in T S * (α; τ ) . The following theorem is about this.
Then the radius of convexity of the function f (z) is By simple computation, we have The last expression is bounded above by 1 if which is equivalent to ∞ ∑ n=2 n 2 a n |z| n−1 ≤ 1. According to Corollary 2.14, Hence, inequality (3.4) will be true if In this section our purpose is to investigate some geometric properties of integral transforms (5.1). We will give the following theorem on the fact that the integral transform F (f, z) of the function f (z) belongs to the class T S * C(α, β; τ ).
Then, by simple computation, we can write It is clear that F ∈ T . Now we need to show that F ∈ T S * C(γ, β; τ ) . For this, according to Theorem 2.7, we need to find the values of γ for which the following inequality is satisfied: (1 − γ) |τ | a n ≤ 1. (1 − α) |τ | a n ≤ 1. which is equivalent to Solving the last inequality for γ , we obtain Thus, the proof of Theorem 5.1 is completed. . The result of the theorem is sharp.