Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli
© Raza and Malik; licensee Springer 2013
Received: 15 February 2013
Accepted: 8 August 2013
Published: 28 August 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
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 .
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 
Lemma 1.2 
Lemma 1.3 
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.
These results are sharp.
Now, using Lemma 1.1, we have the required result. □
where with .
For , we have .
for . Therefore is a point of maximum. Hence, we get the required result. □
These estimations are sharp. The first three bounds were obtained by Sokół and the bound for can be obtained in a similar way.
The authors would like to thank the referees for helpful comments and suggestions which improved the presentation of the paper.
- Sokół J, Stankiewicz J: Radius of convexity of some subclasses of strongly starlike functions. Zesz. Nauk. Politech. Rzeszowskiej Mat. 1996, 19: 101–105.MathSciNetMATHGoogle Scholar
- 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.MathSciNetMATHGoogle Scholar
- Sokół J: Coefficient estimates in a class of strongly starlike functions. Kyungpook Math. J. 2009, 49(2):349–353. 10.5666/KMJ.2009.49.2.349MathSciNetView ArticleMATHGoogle Scholar
- Noonan JW, Thomas DK: On the second Hankel determinant of areally mean p -valent functions. Trans. Am. Math. Soc. 1976, 223(2):337–346.MathSciNetMATHGoogle Scholar
- Dienes P: The Taylor Series. Dover, New York; 1957.MATHGoogle Scholar
- Edrei A: Sur les déterminants récurrents et les singularités d’une fonction donée por son développement de Taylor. Compos. Math. 1940, 7: 20–88.MathSciNetMATHGoogle Scholar
- Cantor DG: Power series with integral coefficients. Bull. Am. Math. Soc. 1963, 69: 362–366. 10.1090/S0002-9904-1963-10923-4MathSciNetView ArticleMATHGoogle Scholar
- Wilson R: Determinantal criteria for meromorphic functions. Proc. Lond. Math. Soc. 1954, 4: 357–374.View ArticleMathSciNetMATHGoogle Scholar
- Vein R, Dale P Applied Mathematical Sciences 134. In Determinants and Their Applications in Mathematical Physics. Springer, New York; 1999.Google Scholar
- Arif M, Noor KI, Raza M: Hankel determinant problem of a subclass of analytic functions. J. Inequal. Appl. 2012. 10.1186/1029-242X-2012-22Google Scholar
- 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 279843Google Scholar
- 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.2012.04.002MathSciNetView ArticleMATHGoogle Scholar
- 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 50Google Scholar
- Janteng A, Halim SA, Darus M: Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 2007, 1(13):619–625.MathSciNetMATHGoogle Scholar
- Babalola KO:On Hankel determinant for some classes of univalent functions. Inequal. Theory Appl. 2007, 6: 1–7.Google Scholar
- 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.Google Scholar
- Grenander U, Szegö G: Toeplitz Forms and Their Applications. University of California Press, Berkeley; 1958.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.