Certain Subclasses of Analytic Functions with Complex Order
Two new subclasses of analytic functions of complex order are introduced. Apart from establishing coefficient bounds for these classes, we establish inclusion relationships involving () neighborhoods of analytic functions with negative coefficients belonging to these subclasses.
Let denote the class of functions of the form which are analytic and univalent in the open disc
A function is star-like of complex order , denoted as if and only if it satisfies
For the two functions given by the Hadamard product or convolution, denoted by , is given by
Given of the form (1) and , we define neighborhood of a function as In particular, for the identity function ,
For complex numbers and (; for ), we define the generalized hypergeometric function as where denotes the set of all positive integers and is the Pochhammer symbol defined in terms of gamma functions as Corresponding to the function defined by recently in , an operator is defined by where and . By the above definition, it is easy to note that Let us take for convenience that Hence, we have For suitable values of , , , , , , and , we can deduce several operators such as Sălăgean derivative operator , Ruscheweyh derivative operator , fractional calculus operator , Carlson-Shaffer operator , Dziok-Srivatsava operator , and also the operator introduced by Abubaker and Darus .
By specializing the parameters involved in the above definitions, we could arrive at several known as well as new classes. For example, by taking , , , , , and and the above classes reduced to where denote the Sălăgean derivative of order given by
Similarly, on taking , , , , , , one gets where is the operator introduced and studied by Abubaker and Darus  given by
Further, by taking in the definition of the classes and , we could arrive at and which were introduced and studied by Murugusundaramoorthy et al. .
In this paper, we establish the coefficient inequalities for the classes and and several inclusion relationships involving neighborhoods of analytic univalent functions with negative and missing coefficients belonging to these classes.
2. Coefficient Inequalities
Theorem 3. Let the function as given in (1). Then, if and only if
Proof. Let the functions of form (1) belong to the class . Then, in view of (15) and (16), we get Letting through real values, we get the required assertion (22). Conversely, suppose satisfies (22), then, in view of (16), consider Hence, the result follows.
Similarly, we prove the following.
Theorem 4. Let the function be as defined in (1). Then, if and only if
Corollary 5. Let the function as given in (1). Then, if and only if
Corollary 6. Let the function be as defined in (1). Then, if and only if
Corollary 7. Let the function as given in (1). Then, if and only if
Corollary 8. Let the function be as defined in (1). Then, if and only if
3. Inclusion Relationships
Theorem 9. If then .
Proof. Let . Then, in view of (22), we have Consider Hence, Hence, the result follows.
In similar manner, we establish the following result.
Theorem 10. If then .
Corollary 11. If then .
Corollary 12. If then .
Corollary 13. If then .
Corollary 14. If then .
4. Neighborhoods for and
In this section, we determine the neighborhood properties of and . Here, the classes consist of functions for which there exists a function such that In the same way, we define , consisting of functions for which there exists another function such that
Theorem 15. If and then .
Proof. Suppose , then
Since , we have Consider Therefore, for given by (42).
Theorem 16. If and then .
Corollary 17. If and then .
Corollary 18. If and then .
Corollary 19. If and then .
Corollary 20. If and then .
Conflict of Interests
The authors declare that they do not have conflict of interests regarding the publication of this paper.
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