On the Inequality Theorem for a Wider Class of Analytic Functions with Hadamard Product

Abstract

In this paper, we discuss a well known class studied by Ramesha in 1995 and later by Janteng in 2006, and we then extend the class to a wider class of functions denoted by which are normalized and univalent, in the unit disk satisfying the condition where is analytic function in , such that , with a new condition that is introduced. The main purpose of this paper is to give an estimate for the same when belongs to the class .

1. Introduction and Definition

Let denote the class of normalized analytic univalent functions of the formwhere . Also, let , , and   be analytic functions in where and  . We define the Hadamard product as follows:

Earlier in 1933, Fekete and Szego [1] states that for and given by (1),for and the inequality is sharp.

The Fekete–Szego problems for the subclass of consisting of the families, convex functions , starlike functions , and close-to-convex functions have been completely solved in the literature. Among others are Keogh and Merkes [2], Koepf [3], Darus and Thomas [4, 5], Frasin and Darus [6], Ebadian et al. [7], and Mohammed et al. [8]. In particular, for and be given by (1), Keogh and Merkes [2] showed thatand for each , there is a function in for which equality holds.

Moreover, an estimate is given for the same functional for the new class defined as follows.

Definition 1. For , , and , let the function be given by (1). Then, the function if and only if there exists analytic function, , such that for ,This class is extended from Ramesha et al. [10] and Janteng [11], for suitable choices of , , we easily obtain the various subclasses of .
For example, if and satisfy (6), we have the class of starlike functions of order denoted by . Also if and satisfy (6), we get the class of convex functions of order denoted by .
To establish our main theorems in this paper, we first state some preliminary lemmas, required in proving of our theorem.

2. Preliminary Results

Lemma 1 (see [9]). Let be analytic in with and be given by

Lemma 2 (see [2]). Let , the starlike function with . Then, for real,The first result for the class is as follows.

Lemma 3. Let the function given by (1) belong to the class . Then,where .

Proof. From (6), we getfor , with given by , where .
From (10), we haveNow, equating coefficients, we getIt also follows from (5) thatwhere , and writing , where , and equating coefficients giveThe result now follows after applying the classical inequalities: , , , and the inequalities and which follow from (12) and (13).
Now we will display the main result for the class .

3. Main Result

Theorem 1. Let the function be given by (1) and belong to the class . Then, for ,where

The inequalities are sharp for all cases. However, the proofs for the case are still unsolved.

Proof. WriteFrom (15) and (16), we have . Then, we getFrom (20), we haveNow, consider the first case for allthat is, having and .
Inequalitiesandfollow immediately from Lemma 1. Now we getNow, after doing some operations, the function achieves its maximum value atNext, we findFinally, we get our results:Now since we know , we get the intervalHence, result (28) concludes for the caseWe shall be certain that the result of this case is sharp. Let , , , , and , in (20).
Secondly, we consider the case .
WriteNow we say that with the aid of the inequality , we get the following:Thus, the proof for this case is complete.
Here, to find we use the previous result of (28), for .
After that, the result is sharp for this case. Upon choosing , and and substituting in (20), we get the result sharp.
Next we consider the case .
The results for this case are correct and sharp, but we have problems to solve for the value . As for , it is very easy and can be directly obtained from the previous result (28).
Without any doubt, we can assure thatwhich is left as conjecture for the readers to solve. Without loss of generality, we would suggest that in the case ,This is sharp on choosing , , , and in (20). Thus, we get the desired equality.
Finally, for the case , we achieve the following easily.
Writeand by using the inequality , we obtain the following:and hence the proof is complete for this case.
HereWe used result (34) forAt last, it remains to show that the result is sharp, upon choosing , , , and and substituting in (20).

4. Conclusion

This article aims at finding a new class of analytic univalent functions on the open unit disc defined by Hadamard product. Further to study their inequality theorem, one of the main requirements needed to satisfy certain classes. This approach, for example, can provide several many fascinating features.

Data Availability

No data were used or available upon request or included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.