Majorization for Certain Classes of Analytic Functions Defined by Fournier–Ruscheweyh Integral Operator

In this paper, we introduce three new classes S ak [ M, N ; μ ] , R ak ( μ ) and T ak ( θ ) of analytic functions defined by Four- nier–Ruscheweyh integral operator. For these classes, we investigate the majorization problem. Furthermore, a number of new results are shown to follow upon specializing the parameters involved in our main results.


Introduction and Definitions
For the two functions u and v which are analytic in the open unit disk D z ∈ C: |z| < 1 { }, we can de ne the majorization for these two functions as follows (see [1]): If there exists a function ψ(z) that is analytic in D, then |ψ(z)| ≤ 1 and u(z) ψ(z)v(z)(z ∈ D).
For the two functions u and v, we say that the function u is subordinate to the function v de ned as u(z)≺v(z), if there is a Schwarz function w, that is analytic in D with w(0) 0 and |w(z)| < 1, z ∈ D, such that u(z) v(w(z)), z ∈ D. Now, on combining subordination and majorization, we de ne quasi-subordination as follows. For two functions u and v, we say that u is quasi-subordinate to v (see [2]) and it is de ned as If there are two functions ψ(z) and w(z) that are analytic in D, then(u(z)/ψ(z)) is analytic in D and |ψ(z)| ≤ 1and w(0) 0, |w(z)| ≤ |z| ≤ 1(z ∈ D), (4) satisfying u(z) ψ(z)v(w(z))(z ∈ D).
Remark 1 (i) If we put ψ(z) 1 in (5), we have the usual denition of subordination (ii) If we put w(z) z in (6) which are analytic in open unit disk D. e function class Φ has been introduced and studied by Li and Srivastava [3] and is de ned as Φ k(t): k(t) ≥ 0, (0 ≤ t ≤ 1), 1 0 k(t)dt 1 .
Fournier and Ruscheweyh [3,4] considered an integral operator with a nonnegative function: By substituting suitable values of parameter a, there are lots of special cases of function k a (t). We therefore consider the Fournier-Ruscheweyh integral operator to be in the following modified form [3] (see [5]): where the real-valued functions k a and k a− 1 fulfill the requirements: (1) For an acceptable parameter a, where t ∈ (0, 1) and − 1 < λ ≤ 2.
For I a k operator, we have Remark 2 in (9), we get the integral operator P a as e integral operator P a is exactly the same as the transformation I k given by Flett [6] and studied subsequently by Li [7], Li and Srivastava [8], and many others. In the case when a > 1, then we have in (9), we get the Jung-Kim-Srivastava integral operator Q a b [9] (see [10][11][12]) as where In terms of known Gamma functions, the integral operator Q a b u(z) is analogous to the convolution operator L(a, d) by Carlson and Shaffer [13]. In the case when a > 1, b > − 1, and 0 < a + b ≤ 3, we have λ � a + b − 1. Now, we describe the following classes of analytical functions using integral operator (9).
If we take the value of k as defined in (13) and (15) where a ≥ 0, k ∈ Φ, and μ ≥ 0. If we take the value of k as defined in (13) and (15), this class becomes R a k 1 (μ) and R a k 2 (μ), respectively.

Problem of Majorization for the Classes
Let the function f ∈ A, and assume that where ρ 0 is the smallest positive root of the equation where w is the analytic function in D, with w(0) � 0 and |w(z)| ≤ |z| < 1∀z ∈ D. Now, from the previous equality, Now, we make use of relation (12), that is, For − 1 < λ ≤ 2, then, from (24), we have which implies that Differentiating the previous equality with respect to z and then multiplying by z, we get On using relation (12), we have is implies Note, therefore, that the Schwarz function ψ satisfies the inequality (see [21]) On using (27) and (32) in (31), we have Setting |z| � ρ and |ψ(z)| � c, then inequality (33) leads to where en, from (34), where From relation (36), in order to prove our result, we need to determine where A simple calculation shows that the G(ρ, c) ≥ 0 inequality is equivalent to However, the function u(ρ, c) has a minimum value at c � 1, that is, Journal of Mathematics λ, M, N) is the smallest positive root of equation (22), which proves conclusion (21).
Proof. Since g ∈ R a k (μ), then, from (19) and the subordination relation, where w is the analytic function in D, with w(0) � 0 and .
In (45), we have which implies that en, we have From (46 and 49), we have Now, on using (12) in (50), for − 1 < λ ≤ 2, we have the following: which implies that Now, since I a k f(z) is majorized by I a k g(z) in D, then we have Differentiating the previous equality with respect to z and then multiplying by z, we get

Corollaries and Consequences
If we take the values of k defined in (13) and (15), then the above theorems give the following corollaries.

Corollary 2.
Let the function f ∈ A, and assume that where ρ 3 is the smallest positive root of the equation where a ≥ 0, μ ≥ 0, and 1 > 2μ + e.

Corollary 4.
Let the function f ∈ A, and assume that where ρ 4 is the smallest positive root of the equation

Corollary 5.
Let the function f ∈ A, and assume that g ∈ R a k 2 (μ).
If we take M � 1 and N � − 1, then eorem 1, Corollary 1, and Corollary 4 give the following results.

Corollary 8.
Let the function f ∈ A, and assume that g ∈ S a k 1 [1, − 1, μ]. If P a f(z) is majorized by P a g(z) in D, then where ρ 2 is the smallest positive root of the equation where μ ∈ C * and 1 ≥ |μ − 1|.

Corollary 9.
Let the function f ∈ A, and assume that g ∈ S a k 2 [1, − 1, μ]. If Q a b f(z) is majorized by Q a b g(z) in D, then where ρ 4 is the smallest positive root of the equation where μ ∈ C * , a > 1, b > − 1, and (a + b) ≥ |2μ − (a + b)|.