On Certain Class of Bazilevič Functions Associated with the Lemniscate of Bernoulli

Abstract

Making use of the principle of subordination, we introduce a certain class of multivalently Bazilevi functions involving the Lemniscate of Bernoulli. Also, we obtain subordination properties, inclusion relationship, convolution result, coefficients estimate, and Fekete–Szegö problem for this class.

1. Introduction

Let be the class of analytic functions in the open unit disk and let denote the subclass of consisting of functions of the form:

We write . For , we say that is subordinate to , written symbolically, in or , if there exists a Schwarz function , which (by definition) is analytic in with and such that . Further more, if the function is univalent in , then we have the following equivalence (see [1, 2]):

Let and be univalent in . If is analytic in and satisfies the first order differential subordination: then is a solution of the differential subordination (4). The univalent function is called a dominant of the solutions of the differential subordination (4) if for all satisfying (4). A univalent dominant that satisfies for all dominants of (4) is called the best dominant.

Sokól and Stankiewicz [3] introduced the class consisting of analytic functions satisfying the following condition which is equivalent to where the function maps onto the domain , and its boundary is the right-half of the lemniscate of Bernoulli . Several geometric properties of were investigated done by many authors in ([4–7]).

Now, we define a class of Bazilevi functions associated with lemniscate of Bernoullia by using the principle of differential subordination as follows.

Definition 1. A function is said to be the class if it satisfies the following subordination condition: all the powers are principal values and throughout the paper unless otherwise mentioned the parameters , , and are constrained as , , , and .

We note that (1)(2) and (3) and (4) and

In order to establish our main results, we need the following lemmas.

Lemma 2 [8]. Let the function be analytic and convex (univalent) in with . Suppose also that the function given by is analytic in . If then and is the best dominant.

Lemma 3. [9]. For real or complex numbers and ,

Lemma 4. [10]. Let be analytic and convex in . If , then

Lemma 5 [11]. Let be analytic in and be analytic and convex in . If then

Lemma 6 [12]. Let , i.e., let be analytic in and satisfy for , then the following sharp estimate holds

The result is sharp for the functions given by

Lemma 7. [12]. If , then when or , the equality holds if and only if or one of its rotations. If , then the equality holds if and only if or one of its rotations. If , the equality holds if and only if or one of its rotations. If , the equality holds if and only if is the reciprocal of one of the functions such that equality holds in the case of .

Also, the above upper bound is sharp, and it can be improved as follows when :

In the present paper, we obtain subordination properties, inclusion relationship, convolution result, coefficients estimate, and Fekete–Szegö inequalities for the class .

2. Main Results

We begin by presenting our first subordination property given by Theorem 8.

Theorem 8. If with , then where the function given by is the best dominant.

Proof. Let and suppose that Then, the function is of the form (9), analytic in , and . By taking the derivatives in the both sides of (22), we get Since , we have Now, by using Lemma 2 for , we deduce that where we have made a change of variables followed by the use of identities in Lemma 3 with , , and . This completes the proof of Theorem 8.

For a function given by (2), the generalized Bernardi-Libera-Livingston integral operator , with , is defined by (see [13–16])

It is easy to verify that for all , we have

Theorem 9. If the function satisfies the subordination condition and is the integral operator defined by (26), then where the function given by is the best dominant of (28).

Proof. Let then is analytic in . Differentiating (31) with respect to and using the identity (28) in the resulting relation, we get Employing the same technique that we used in the proof of Theorem 8, the remaining part of the theorem can be proved similarly.

Theorem 10. If , then

Proof. Suppose that . We know that Thus, the assertion of Theorem 10 holds for . If , by Theorem 8 and (34), we have At the same time, we have Moreover, since and the function is analytic and convex in .
Combining (34)–(36) and Lemma 4, we find that that is , which implies that the assertion (33) of Theorem 10 holds.

Theorem 11. If , then if and only if where

Proof. For any function we can verify that First, in order to prove that (38) holds, we will write (8) by using the principle of subordination, that is, where is a Schwarz function, hence for all and . From (40) and (41), the relation (43) may be written as which is equivalent to that is (38).
Reversely, suppose that satisfy the condition (38). Like it was previously shown, the assumption (38) is equivalent to (41), that is, Denoting the relation (46) could be written as . Therefore, the simply connected domain is included in a connected component of . From this fact, using that together with the univalence of the function , it follows that , that is .

Theorem 12. If given by (2) belongs to , then

Proof. Combining (2) and (8), we obtain An application of Lemma 5 to (49) yields Thus, from (50), we easily obtain (48) asserted by Theorem 12.

3. Fekete–Szegö Problem

Many authors have considered the Fekete–Szegö problem for many subclasses of analytic functions (see, for instance, [17–21]). In this section, we evaluate the Fekete–Szegö inequalities for the class .

Theorem 13. If given by (2) belongs to the class , then

The result is sharp.

Proof. If , then there is a Schwarz function in such that where . Define the function by Since is a Schwarz function, we see that with . Therefore, Now by substituting (54) in (52), we have Equating the coefficients of and , we obtain Therefore, where Our result now follows by an application of Lemma 6. The result is sharp for the functions This completes the proof of Theorem 13.

Putting and in Theorem 13, we obtain the following corollary.

Corollary 14. If given by (2) belongs to the class , then

The result is sharp.

Putting and in Theorem 13, we obtain the following corollary.

Corollary 15. If given by (2) belongs to the class , then

The result is sharp.

Putting and in Theorem 13, we obtain the following corollary.

Corollary 16. If given by (2) (with ) belongs to the class , then

The result is sharp.

Applying Lemma 7 to (57) and (58), we obtain the following theorem.

Theorem 17. Let

If given by (2) belongs to the class , then

Further, if , then

If , then

Putting in Theorem 17, we obtain the following result.

Corollary 18. Let

If given by (2) belongs to the class , then

Further, if , then

If , then

Putting and in Theorem 17, we obtain the following result for the subclass .

Corollary 19. If given by (2) belongs to the class , then

Further, if , then

If , then

Putting and in Theorem 17, we obtain the following result obtained by ([18], Theorem 2.1).

Corollary 20. ([18], Theorem 2.1). If given by (2) (with ) belongs to the class , then

Further, if , then

If , then

Data Availability

No data were used to support this study.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.