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

Abstract

In this paper, we introduce three new classes , and of analytic functions defined by Fournier–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.

1. Introduction and Definitions

For the two functions and which are analytic in the open unit disk , we can define the majorization for these two functions as follows (see [1]):

If there exists a function that is analytic in , then

For the two functions and , we say that the function is subordinate to the function defined as , if there is a Schwarz function , that is analytic in with and , , such that , .

Now, on combining subordination and majorization, we define quasi-subordination as follows. For two functions and , we say that is quasi-subordinate to (see [2]) and it is defined as

If there are two functions and that are analytic in , then() is analytic in andsatisfying

Remark 1. (i)If we put in (5), we have the usual definition of subordination(ii)If we put in (6), we have the usual definition of majorizationLet denote the class of all functions of the formwhich are analytic in open unit disk .
The function class has been introduced and studied by Li and Srivastava [3] and is defined asFournier and Ruscheweyh [3, 4] considered an integral operator with a nonnegative function:By substituting suitable values of parameter , there are lots of special cases of function . We therefore consider the Fournier–Ruscheweyh integral operator to be in the following modified form [3] (see [5]):where the real-valued functions and fulfill the requirements:(1)For an acceptable parameter ,(2)There exists a constant such that where and .For operator, we have

Remark 2. (i)If we take in (9), we get the integral operator as The integral operator is exactly the same as the transformation given by Flett [6] and studied subsequently by Li [7], Li and Srivastava [8], and many others. In the case when , then we have .(ii)If we take in (9), we get the Jung–Kim–Srivastava integral operator [9] (see [10–12]) aswhereIn terms of known Gamma functions, the integral operator is analogous to the convolution operator by Carlson and Shaffer [13]. In the case when , and , we have .
Now, we describe the following classes of analytical functions using integral operator (9).

Definition 1. The function is said to be in the class if and only ifwith .
If we take the value of as defined in (13) and (15), this class becomes and , respectively.

Definition 2. The function is said to be in the class if and only ifwhere , and .
If we take the value of as defined in (13) and (15), this class becomes and , respectively.

Definition 3. The function is said to be in the class if and only ifwhere , and .
If we take the value of as defined in (13) and (15), this class becomes and , respectively.
A majorization problem for the normalized class of starlike functions has been investigated by MacGregor [1] and further studied by Altintas et al. [14]. Recently, a number of researchers have studied several majorization problems for univalent and multivalent functions or meromorphic and multivalent meromorphic functions involving different operators and different classes [14–20, 22–24]. By motivating the above work, the majorization problems of the classes , and are investigated as follows.

2. Problem of Majorization for the Classes , , and

Theorem 1. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

Proof. Since , then, from (18),where is the analytic function in , with and .
Now, from the previous equality,Now, we make use of relation (12), that is,For , then, from (24), we havewhich implies thatNow, since is majorized by in open unit disk , thenDifferentiating the previous equality with respect to and then multiplying by , we getOn using relation (12), we haveThis impliesNote, therefore, that the Schwarz function satisfies the inequality (see [21])On using (27) and (32) in (31), we haveSetting , then inequality (33) leads towhereThen, from (34),whereFrom relation (36), in order to prove our result, we need to determinewhereA simple calculation shows that the inequality is equivalent toHowever, the function has a minimum value at , that is,whereIt follows that , where is the smallest positive root of equation (22), which proves conclusion (21).

Theorem 2. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

Proof. Since then, from (19) and the subordination relation,where is the analytic function in , with and . Now, letIn (45), we havewhich implies thatThen, we haveFrom (46 and 49), we haveNow, on using (12) in (50), for , we have the following:which implies thatNow, since is majorized by in , then we haveDifferentiating the previous equality with respect to and then multiplying by , we getOn using relation (12), we haveThis impliesThus, note that the Schwarz function satisfies the inequality (see [21])On using (52) and (57) in (56), we haveSetting , then inequality (58) leads towhereThen, from (59),Here,From relation (61), in order to prove our result, we need to determinewhereA simple calculus shows that the inequality is equivalent toHowever, the function takes its minimum value at , that is,whereIt follows that , where is the smallest positive root of (44), which prove conclusion (43).

Theorem 3. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

Proof. Since , then, from (20) and the subordination relation,where is the analytic function in , with and . From (70), we haveNow, on using (12) in (71), for , we have the following:which implies thatNow, since is majorized by in , then we haveDifferentiating the previous equality with respect to and then multiplying by , we getOn using relation (12), we haveThis impliesThus, note that the Schwarz function satisfies the inequality (see [21])On using (71) and (78) in (77), we haveSetting , then inequality (79) leads towhereThen, from (80),whereFrom relation (82), in order to prove our result, we need to determinewhereA simple calculus shows that the inequality is equivalent toHowever, the function takes its minimum value at , that is,whereIt follows that , where is the smallest positive root of (69), which proves conclusion (68).

3. Corollaries and Consequences

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

Corollary 1. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

Corollary 2. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

Corollary 3. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

Corollary 4. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

Corollary 5. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

Corollary 6. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

If we take and , then Theorem 1, Corollary 1, and Corollary 4 give the following results.

Corollary 7. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

Corollary 8. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere and .

Corollary 9. Let the function , and assume that . If is majorized by in , thenwhere is the smallest positive root of the equationwhere , and .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors have contributed equally to the paper.

Acknowledgments

This work was supported by the Taif University Researchers Supporting Project (TURSP-2020/305), Taif University, Taif, Saudi Arabia.