Abstract
Keywords:
starlike functions; subordination; lemniscate of Bernoulli; Toeplitz determinants; Hankel determinants1 Introduction and preliminaries
Let A be the class of functions f of the form
which are analytic in the open unit disk . A function f is said to be subordinate to a function g, written as , if there exists a Schwartz function w with and such that . In particular, if g is univalent in E, then and .
Let P denote the class of analytic functions p normalized by
such that . Let be the class of functions defined by
Thus a function is such that lies in the region bounded by the right half of the lemniscate of Bernoulli given by the relation . It can easily be seen that if it satisfies the condition
This class of functions was introduced by Sokół and Stankiewicz [1] and further investigated by some authors. For details, see [2,3].
Noonan and Thomas [4] have studied the qth Hankel determinant defined as
where and . The Hankel determinant plays an important role in the study of singularities; for instance, see [[5], p.329] and Edrei [6]. This is also important in the study of power series with integral coefficients [[5], p.323] and Cantor [7]. For the use of the Hankel determinant in the study of meromorphic functions, see [8], and various properties of these determinants can be found in [[9], Chapter 4]. It is well known that the FeketeSzegö functional . This functional is further generalized as for some μ (real as well as complex). Fekete and Szegö gave sharp estimates of for μ real and , the class of univalent functions. It is a very great combination of the two coefficients which describes the area problems posted earlier by Gronwall in 191415. Moreover, we also know that the functional is equivalent to . The qth Hankel determinant for some subclasses of analytic functions was recently studied by Arif et al.[10] and Arif et al.[11]. The functional has been studied by many authors, see [1214]. Babalola [15] studied the Hankel determinant for some subclasses of analytic functions. In the present investigation, we determine the upper bounds of the Hankel determinant for a subclass of analytic functions related with lemniscate of Bernoulli by using Toeplitz determinants.
We need the following lemmas which will be used in our main results.
Lemma 1.1[16]
Letand of the form (1.2). Then
Whenor, the equality holds if and only ifisor one of its rotations. If, then the equality holds if and only ifor one of its rotations. If, the equality holds if and only if () or one of its rotations. If, the equality holds if and only ifpis the reciprocal of one of the functions such that the equality holds in the case of. Although the above upper bound is sharp, when, it can improved as follows:
and
Lemma 1.2[16]
Ifis a function with positive real part inE, then forva complex number
This result is sharp for the functions
Lemma 1.3[17]
Letand of the form (1.2). Then
2 Main results
Although we have discussed the Hankel determinant problem in the paper, the first two problems are specifically related with the FeketeSzegö functional, which is a special case of the Hankel determinant.
Theorem 2.1Letand of the form (1.1). Then
These results are sharp.
Proof If , then it follows from (1.3) that
It is clear that . This implies that
From (2.1), we have
with
Now
Similarly,
Therefore
This implies that
Now, using Lemma 1.1, we have the required result. □
The results are sharp for the functions , , such that
Theorem 2.2Letand of the form (1.1). Then for a complex numberμ,
Proof Since
therefore, using Lemma 1.2, we get the result. This result is sharp for the functions
or
□
Corollary 2.3Letand of the form (1.1). Then
Theorem 2.4Letand of the form (1.1). Then
Proof From (2.2), (2.3) and (2.4), we obtain
Putting the values of and from Lemma 1.3, we assume that , and taking , we get
After simple calculations, we get
Now, applying the triangle inequality and replacing by ρ, we obtain
Differentiating with respect to ρ, we have
It is clear that , which shows that is an increasing function on the closed interval . This implies that maximum occurs at . Therefore (say). Now
Therefore
and
for . This shows that maximum of occurs at . Hence, we obtain
This result is sharp for the functions
or
□
Theorem 2.5Letand of the form (1.1). Then
Proof Since
Therefore, by using Lemma 1.3, we can obtain
Let
We assume that the upper bound occurs at the interior point of the rectangle . Differentiating (2.5) with respect to ρ, we get
For and fixed , it can easily be seen that . This shows that is a decreasing function of ρ, which contradicts our assumption; therefore, . This implies that
and
for . Therefore is a point of maximum. Hence, we get the required result. □
Lemma 2.6If the functionbelongs to the class, then
These estimations are sharp. The first three bounds were obtained by Sokół[3]and the bound forcan be obtained in a similar way.
Theorem 2.7Letand of the form (1.1). Then
Proof Since
Now, using the triangle inequality, we obtain
Using the fact that with the results of Corollary 2.3, Theorem 2.4, Theorem 2.5 and Lemma 2.6, we obtain
□
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MR and SNM jointly discussed and presented the ideas of this article. MR made the text file and all the communications regarding the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for helpful comments and suggestions which improved the presentation of the paper.
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