- Open Access
Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli
Journal of Inequalities and Applications volume 2013, Article number: 412 (2013)
In this paper, the upper bound of the Hankel determinant for a subclass of analytic functions associated with right half of the lemniscate of Bernoulli is investigated.
1 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
Noonan and Thomas  have studied the q th Hankel determinant defined as
where and . The Hankel determinant plays an important role in the study of singularities; for instance, see [, p.329] and Edrei . This is also important in the study of power series with integral coefficients [, p.323] and Cantor . For the use of the Hankel determinant in the study of meromorphic functions, see , and various properties of these determinants can be found in [, Chapter 4]. It is well known that the Fekete-Szegö 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 1914-15. Moreover, we also know that the functional is equivalent to . The q th Hankel determinant for some subclasses of analytic functions was recently studied by Arif et al.  and Arif et al. . The functional has been studied by many authors, see [12–14]. Babalola  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 
Let and of the form (1.2). Then
When or , the equality holds if and only if is 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 p is 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:
Lemma 1.2 
If is a function with positive real part in E, then for v a complex number
This result is sharp for the functions
Lemma 1.3 
Let and of the form (1.2). Then
for some x, , and
for some z, .
2 Main results
Although we have discussed the Hankel determinant problem in the paper, the first two problems are specifically related with the Fekete-Szegö functional, which is a special case of the Hankel determinant.
Theorem 2.1 Let and of the form (1.1). Then
Furthermore, for ,
and for ,
These results are sharp.
Proof If , then it follows from (1.3) that
where . Define a function
It is clear that . This implies that
From (2.1), we have
This implies that
Now, using Lemma 1.1, we have the required result. □
The results are sharp for the functions , , such that
where with .
Theorem 2.2 Let and of the form (1.1). Then for a complex number μ,
therefore, using Lemma 1.2, we get the result. This result is sharp for the functions
For , we have .
Corollary 2.3 Let and of the form (1.1). Then
Theorem 2.4 Let and 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
for . This shows that maximum of occurs at . Hence, we obtain
This result is sharp for the functions
Theorem 2.5 Let and of the form (1.1). Then
Therefore, by using Lemma 1.3, we can obtain
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
for . Therefore is a point of maximum. Hence, we get the required result. □
Lemma 2.6 If the function belongs to the class , then
These estimations are sharp. The first three bounds were obtained by Sokół and the bound for can be obtained in a similar way.
Theorem 2.7 Let and of the form (1.1). Then
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
Sokół J, Stankiewicz J: Radius of convexity of some subclasses of strongly starlike functions. Zesz. Nauk. Politech. Rzeszowskiej Mat. 1996, 19: 101–105.
Ali RM, Cho NE, Ravichandran V, Kumar SS: Differential subordination for functions associated with the lemniscate of Bernoulli. Taiwan. J. Math. 2012, 16(3):1017–1026.
Sokół J: Coefficient estimates in a class of strongly starlike functions. Kyungpook Math. J. 2009, 49(2):349–353. 10.5666/KMJ.2009.49.2.349
Noonan JW, Thomas DK: On the second Hankel determinant of areally mean p -valent functions. Trans. Am. Math. Soc. 1976, 223(2):337–346.
Dienes P: The Taylor Series. Dover, New York; 1957.
Edrei A: Sur les déterminants récurrents et les singularités d’une fonction donée por son développement de Taylor. Compos. Math. 1940, 7: 20–88.
Cantor DG: Power series with integral coefficients. Bull. Am. Math. Soc. 1963, 69: 362–366. 10.1090/S0002-9904-1963-10923-4
Wilson R: Determinantal criteria for meromorphic functions. Proc. Lond. Math. Soc. 1954, 4: 357–374.
Vein R, Dale P Applied Mathematical Sciences 134. In Determinants and Their Applications in Mathematical Physics. Springer, New York; 1999.
Arif M, Noor KI, Raza M: Hankel determinant problem of a subclass of analytic functions. J. Inequal. Appl. 2012. 10.1186/1029-242X-2012-22
Arif M, Noor KI, Raza M, Haq SW: Some properties of a generalized class of analytic functions related with Janowski functions. Abstr. Appl. Anal. 2012., 2012: Article ID 279843
Bansal D: Upper bound of second Hankel determinant for a new class of analytic functions. Appl. Math. Lett. 2013, 26(1):103–107. 10.1016/j.aml.2012.04.002
Janteng A, Halim SA, Darus M: Coefficient inequality for a function whose derivative has a positive real part. J. Inequal. Pure Appl. Math. 2006., 7(2): Article ID 50
Janteng A, Halim SA, Darus M: Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 2007, 1(13):619–625.
Babalola KO:On Hankel determinant for some classes of univalent functions. Inequal. Theory Appl. 2007, 6: 1–7.
Ma WC, Minda D: A unified treatment of some special classes of univalent functions. In Proceedings of the Conference on Complex Analysis. Edited by: Li Z, Ren F, Yang L, Zhang S. Int. Press, Cambridge; 1994:157–169.
Grenander U, Szegö G: Toeplitz Forms and Their Applications. University of California Press, Berkeley; 1958.
The authors would like to thank the referees for helpful comments and suggestions which improved the presentation of the paper.
The authors declare that they have no competing interests.
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.
About this article
Cite this article
Raza, M., Malik, S.N. Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli. J Inequal Appl 2013, 412 (2013) doi:10.1186/1029-242X-2013-412
- starlike functions
- lemniscate of Bernoulli
- Toeplitz determinants
- Hankel determinants