Abstract
Making use of a differential operator, which is defined here by means of the Hadamard product, we introduce classes of -valent functions and investigate various important inclusion properties and characteristics for these classes. Also, a property preserving integrals is considered.
1. Introduction
Let be the class of functions which are analytic and -valent in .
If and are analytic in , is subordinate to , if there exists an analytic function and such that . Furthermore, if is univalent in , then (see [1, 2])
For functions , given by (1) and defined by the Hadamard product of and is given by
For , denote by and the classes of -valently starlike and convex functions of order and , respectively (see [3, 4]), satisfying
It follows from (5) and (6) that
See Goodman [5].
Also, denote by and the classes of -valently close-to-convex and quasi-convex functions of order and type satisfying, respectively (see [6–8] (with ),
It follows from (8) and (9) that
Dziok and Srivastava [9] used the hypergeometric function (see Srivastava and Karlsson [10]) and defined the linear operator where
Setting the function we define a function in terms of the Hadamard product (or convolution) by
Let
From (16), it can be easy to verify that
Using the operator , we introduce the subclasses.
We note that
2. Main Results
Unless otherwise mentioned, we assume that and .
The following lemma due to Miller and Mocanu is required to prove the results.
Lemma 1 (see [11]). Let be the complex function being the complex plane and let , . Suppose that satisfies the following conditions: (i) is continuous in (ii) and (iii) for all and such that Let be regular in such that for all . If then
Theorem 2. Let . Then,
Proof. Let , and where given by (26) we have Differentiating (31), we have which, in view of (30), leads to Let with . Then, (i) is continuous in (ii) and (iii) for all and such that for ; therefore, the function satisfies the conditions in Lemma 1; thus, we have , that is, .
Theorem 3. For , we have
Proof. Applying (23) and using Theorem 2, we have
Theorem 4. For , we have
Proof. Let ; then, from (21), there exists a function such that
Put
where is given by (26). Applying (17) in (40) differentiating the resulting equation and multiplying by , we have
Since , then by Theorem 2, we have . Let
where . Applying (17) in (42), we have
From (41) and (43), we have
Now, let
, with . Then,
(i) is continuous in (ii) and (iii) for all and such that for , where , and being functions of and , and ; thus, we have , that is, .
Theorem 5. For , we have
Proof. Using (24), we can prove Theorem 5 as that making in Theorem 3.
3. Inclusion Results for
The generalized Libera integral operator (see [12]) is defined by which satisfies
Theorem 6. Let and ; then, .
Proof. Let and put where is given by (26). Applying (48) in (49), we have Differentiating (50), we have where in view of (49), we have Let with . Then, (i) is continuous in (ii) and (iii) for all and such that for ; therefore, the function satisfies the conditions in Lemma 1, and .
Theorem 7. Let and ; then, .
Proof. Applying Theorem 6 and (23), we have
Theorem 8. Let and ; then, .
Proof. Let ; then, from (21), there exists a function such that Put where is given by (26). Applying (48) in (57) differentiating the resulting equation with respect to and multiplying by , we have Since , then by Theorem 6, we have . Let where . Applying (48) in (59), we have From (58) and (60), we have Now, let It is easy to see that satisfies the conditions (i) and (ii) of Lemma 1 in . For (iii), we have for , where , and being functions of and , and Thus, we have , that is, .
Similarly, we can prove the following theorem.
Theorem 9. Let and ; then, .
Remark 10. (1)Using (18) instead of (17) in Theorems 2–5, we have new inclusion results(2)For special values of the parameters in (16), we obtain another new inclusion results for different classes
4. Conclusion
Using the hypergeometric function (see Srivastava and Karlsson [10]) and Hadamard product, we defined an operator for -valent functions. This operator generalizes many other operators for special values of its parameters and has two recurrence relations and then defined four classes related to starlike, convex, close-to-convex, and quasi-to-convex -valent functions. We used Miller and Mocanu lemma [11] for second differential inequalities to obtain inclusion relations for these classes and also for the generalized Libera integral operator.
5. Future Studies
The authors suggest to obtain the inclusion results for the classes using the following lemma according to Jack [13] instead of Lemma 1.
Jack’s lemma [13] state that, if is analytic function in , with , attains its maximum value on the circle at a point and ; then,
Data Availability
During the current study, the data are derived arithmetically.
Conflicts of Interest
The authors do not have any competing interests.
Authors’ Contributions
The authors approve and read the article.