Some Applications of Second-Order Differential Subordination on a Class of Analytic Functions Defined by Komatu Integral Operator
We introduce a new class of analytic functions by using Komatu integral operator and obtain some subordination results.
1. Introduction, Definitions, and Preliminaries
Let be the set of real numbers, the set of complex numbers, be the set of positive integers, and
Let be the class of analytic functions in the open unit disk and the subclass of consisting of the functions of the form
Let be the class of all functions of the form which are analytic in the open unit disk with Also let denote the subclass of consisting of functions which are univalent in .
A function analytic in is said to be convex if it is univalent and is convex.
Let denote the class of normalized convex functions in .
If and are analytic in , then we say that is subordinate to , written symbolically as if there exists a Schwarz function which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence [1, page 4]:
Let be a function and let be univalent in . If is analytic in and satisfies the (second-order) differential subordination then is called a solution of the differential subordination. The univalent function is called a dominant of the solutions of the differential subordination, or more simply a dominant, if for all satisfying (13).
A dominant , which satisfies for all dominants of (13), is said to be the best dominant of (13).
Recently, Komatu  introduced a certain integral operator defined by
Thus, if is of the form (5), then it is easily seen from (14) that (see )
Using the relation (15), it is easy verify that
We note the following.(i)For and ( is any integer), the multiplier transformation was studied by Flett  and Sălăgean .(ii)For and (), the differential operator was studied by Sălăgean .(iii)For and ( is any integer), the operator was studied by Uralegaddi and Somanatha .(iv)For , the multiplier transformation was studied by Jung et al. .
Using the operator , we now introduce the following class.
Definition 1. Let be the class of functions satisfying where , , and is the Komatu integral operator.
In order to prove our main results, we will make use of the following lemmas.
Lemma 2 (see ). Let be a convex function with and let be a complex number with . If and then where The function is convex and is the best dominant.
Lemma 3 (see ). Let , , and let Let be an analytic function in with and suppose that If is analytic in and then where is a solution of the differential equation given by Moreover is the best dominant.
2. Main Results
Theorem 4. The set is convex.
Proof. Let be in the class . Then, by Definition 1, we have For any nonnegative numbers such that we must show that the function is in ; that is, By (28) and (31), we have Therefore we get Differentiating (34) with respect to , we obtain So we get since . Therefore we get the desired result.
Theorem 5. Let be convex function in with and let where is a complex number with . If and , where then implies and this result is sharp.
Proof. From the equality (38), we get Differentiating (41) with respect to , we have Differentiating (43) with respect to , we obtain Using the differential subordination (39) in the equality (44), we get Let us define Then a simple computation yields Using (46) in the subordination (45), we have Using Lemma 2, we obtain which is desired result. Moreover is the best dominant.
Example 6. If we take in Theorem 5, then we have If and is given by then by Theorem 5, we have
Theorem 7. Let and let Let be an analytic function in with and suppose that If and , where is defined by (38), then implies where is the solution of the differential equation given by Moreover is the best dominant.
Proof. We consider and in Lemma 3. Then the proof is easily seen by means of the proof of Theorem 5.
Letting in Theorem 7, we obtain the following interesting result.
Corollary 8. If , , , , and is defined by (38), then where and this results is sharp. Moreover where
Proof. If we let then is convex and by Theorem 7, we deduce On the other hand if , then from the convexity of and the fact that is symmetric with respect to the real axis, we get where is given by (64). From (66), we have where is given by (63).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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