Estimate for Initial MacLaurin Coefficients of Certain Subclasses of Bi-univalent Functions of Complex Order Associated with the Hohlov Operator

In this paper we introduce and investigate two new subclasses of the function class $\Sigma$ of bi-univalent functions of complex order defined in the open unit disk, which are associated with the Hohlov operator, and satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-MacLaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclasses. Several known or new consequences of these results are also pointed out.


Introduction
Let A denote the class of functions of the form f (z) = z + ∞ n=2 a n z n , (1.1) which are analytic in the open unit disk U := {z ∈ C : |z| < 1}. By S we will denote the subclass of all functions in A which are univalent in U. Some of the important and well-investigated subclasses of the class S include, for example, the class S * (α) of starlike functions of order α in U, and the class K(α) of convex functions of order α in U, with 0 ≤ α < 1.

(1.2)
A function f ∈ A is said to be bi-univalent in U if f (z) and f −1 (w) are univalent in U, and let Σ denote the class of bi-univalent functions in U.
The convolution or Hadamard product of two functions f, h ∈ A is denoted by f * h, and is defined by where f is given by (1.1) and h(z) = z + ∞ n=2 b n z n . Next, in our present investigation, we need to recall the convolution operator I a,b,c due to Hohlov [11,10], which is a special case of the Dziok-Srivastava operator [5,6].
For the complex parameters a, b and c (c = 0, −1, −2, −3, . . . ), the Gaussian hypergeometric function 2 F 1 (a, b, c; z) is defined as where (α) n is the Pochhammer symbol (or the shifted factorial) given by For the real positive values a, b and c, using the Gaussian hypergeometric function (1.3), Hohlov [11,10] introduced the familiar convolution operator I a,b,c : A → A by 5) and the function f is of the form (1.1). Hohlov [11,10] discussed some interesting geometrical properties exhibited by the operator I a,b,c , and the three-parameter family of operators I a,b,c contains, as its special cases, most of the known linear integral or differential operators. In particular, if b = 1 in (1.4), then I a,b,c reduces to the Carlson-Shaffer operator. Similarly, it is easily seen that the Hohlov operator I a,b,c is also a generalization of the Ruscheweyh derivative operator as well as the Bernardi-Libera-Livingston operator. It is of interest to note that for a = c and b = 1, then I a,1,a f = f , for all f ∈ A.
Recently there has been triggering interest to study bi-univalent function class Σ and obtained non-sharp coefficient estimates on the first two coefficients |a 2 | and |a 3 | of (1.1). But the coefficient problem for each of the following Taylor-MacLaurin coefficients is still an open problem (see [2,1,3,12,14,20]). Many researchers (see [7,9,13,18]) have recently introduced and investigated several interesting subclasses of the bi-univalent function class Σ and they have found non-sharp estimates on the first two Taylor-MacLaurin coefficients |a 2 | and |a 3 |.
For β = 0, we denote P m := P m (0), hence the class P m represents the class of functions p analytic in U, normalized with p(0) = 1, and having the representation where µ is a real-valued function with bounded variation, which satisfies Motivated by the earlier work of Deniz [4], Peng et al. [17] (see also [16,19]) and Goswami et al. [8], in the present paper we introduce new subclasses of the function class Σ of complex order γ ∈ C * := C \ {0}, involving Hohlov operator I a,b,c , and we find estimates on the coefficients |a 2 | and |a 3 | for the functions that belong to these new subclasses of functions of the class Σ. Several related classes are also considered, and connection to earlier known results are made.
if the following two conditions are satisfied: where γ ∈ C * , the function g is given by (1.2), and z, w ∈ U.
Definition 2.3. For 0 ≤ λ ≤ 1 and 0 ≤ β < 1, a function f ∈ Σ is said to be in the class K a,b,c Σ (γ, λ, β) if it satisfies the following two conditions: where γ ∈ C * , the function g is given by (1.2), and z, w ∈ U.
On specializing the parameters λ one can state the various new subclasses of Σ as illustrated in the following examples. Thus, taking λ = 1 in the above two definitions, we obtain: (i) A function f ∈ Σ is said to be in the class S a,b,c Σ (γ, β) if the following conditions are satisfied: where g = f −1 and z, w ∈ U.
(ii) A function f ∈ Σ is said to be in the class K a,b,c Σ (γ, β) if it satisfies the following conditions: where g = f −1 and z, w ∈ U.
Taking λ = 0 in the previous two definitions, we obtain the next special cases: if the following conditions are satisfied: where g = f −1 and z, w ∈ U.
(ii) A function f ∈ Σ is said to be in the class Q a,b,c Σ (γ, β) if it satisfies the following conditions: where g = f −1 and z, w ∈ U.
In particular, for a = c and b = 1, we note that I a,1,a f = f for all f ∈ A, and thus, for λ = 1 and λ = 0 the classes S a,b,c Σ (γ, λ, β) and K a,b,c Σ (γ, λ, β) reduces to the following subclasses of Σ, respectively: Example 2.3. (i) For 0 ≤ β < 1 and γ ∈ C * , a function f ∈ Σ is said to be in the class S * Σ (γ, β) if the following conditions are satisfied: where g = f −1 and z, w ∈ U.
(ii) For 0 ≤ β < 1 and γ ∈ C * , a function f ∈ Σ is said to be in the class K Σ (γ, β) if the following conditions are satisfied: where g = f −1 and z, w ∈ U.
Example 2.4. (i) For 0 ≤ β < 1 and γ ∈ C * , a function f ∈ Σ is said to be in the class H Σ (γ, β) if the following conditions are satisfied: where g = f −1 and z, w ∈ U.
(ii) For 0 ≤ β < 1 and γ ∈ C * , a function f ∈ Σ is said to be in the class Q Σ (γ, β) if the following conditions are satisfied: where g = f −1 and z, w ∈ U.
In order to derive our main results, we shall need the following lemma: h n z n , z ∈ U, such that Φ ∈ P m (β).
By employing the techniques used earlier by Deniz [4], in the following section we find estimates of the coefficients |a 2 | and |a 3 | for functions of the above-defined subclasses S a,b,c Σ (γ, λ, β) and K a,b,c Σ (γ, λ, β) of the function class Σ.
3. Coefficient Bounds for the Function Class S a,b,c Σ (γ, λ, β) We begin by finding the estimates on the coefficients |a 2 | and |a 3 | for functions belonging to the class S a,b,c Σ (γ, λ, β). Supposing that the functions p, q ∈ P m (β), with from Lemma 2.1 it follows that and where ϕ 2 and ϕ 3 are given by (1.5).
For the special cases λ = 1 and λ = 0, the Theorem 3.1 reduces to the following corollaries, respectively: If the function f given by (1.1) belongs to the class S a,b,c Σ (γ, β), then where ϕ 2 and ϕ 3 are given by (1.5).
Remark 4.1. For a = c and b = 1, we have ϕ n = 1 for all n ≥ 1, and taking γ = 1 and m = 2 in Corollary 3.1 and Corollary 3.2 we obtain more accurate results corresponding to the results obtained in [19,18]. (ii) Moreover, the operators I 1,1,2 and I 1,2,3 are the well-known Alexander and Libera operators, respectively.
(iii) Further, if we take b = 1 in (1.4), then I a,1,c immediately yields the Carlson-Shaffer operator, that is L(a, c) := I a,1,c .
Remark that, various other interesting corollaries and consequences of our main results, which are asserted by Theorem 3.1 and Theorem 4.1 above, can be derived similarly. The details involved may be left as exercises for the interested reader.