Hankel determinant problem of a subclass of analytic functions
© Arif et al; licensee Springer. 2012
Received: 2 October 2011
Accepted: 8 February 2012
Published: 8 February 2012
In this article, we study the Hankel determinant problem of a subclass of analytic functions introduced recently by Arif et al.
2010 Mathematics Subject Classification: 30C45; 30C10.
Let be the class of analytic function satisfying the condition f (0) = 0, f'(0) - 1 = 0 in the open unit disc . By , , , and we means the well-known subclasses of which consist of univalent, starlike, convex, and close-to-convex functions, respectively.
We now consider a class of analytic functions defined by Arif et al.  as follows:
where and p(z) belongs to the class of functions whose real part is positive.
Throughout in this article, we shall assume, unless otherwise stated, that k ≥ 2, 0 ≤ β ≤ 1, 0 < γ ≤ 1, λ is real with , 0 ≤ σ < 1.
In , the q th Hankel determinant , q ≥ 1, n ≥ 1, for a function is stated by Noonan and Thomas as:
The Hankel determinant plays an important role, for instance, in the study of the singularities by Hadamard, see [, p. 329], Edrei  and in the study of power series with integral coefficients by Polya [, p. 323], Cantor , and many others.
In this article, we shall determine the rate of growth of the Hankel determinant for with 0 < β < 2, as n → ∞. This determinant has been considered by several authors. That is, Noor  determined the rate of growth of as n → ∞ for a function f(z) belongs to the class . Pommerenke in  studied the Hankel determinant for starlike functions. The Hankel determinant problem for other interesting classes of analytic functions were discussed by Noor [11, 12, 24].
Lemmas 1.1 and 1.2 are due to Noonan and Thomas .
there exists a z1 with |z1| = r such that for all z, |z| = r
2 Hankel determinant problem
where and O(1) is a constant depending on k, λ, β, σ, γ, and j only.
where c2 is a constant depending on k, λ, β, σ, γ, j only and γ(k + 2) (1 - σ) cos2 λ - 4j > 2 - β.
where O(1) is a constant depending on k, λ, β, σ, γ, and j only.
We now estimate the rate of growth of .
and O(1) depends only on k, λ, β, σ, γ, and j.
where O(1) is a constant depending only on k and γ.
where O(1) is a constant depends on k, σ, and β only.
- Moulis EJ: Generalizations of the Robertson functions. Pac J Math 1979, 81(1):167–174.MATHMathSciNetView ArticleGoogle Scholar
- Padmanabhan K, Parvatham R: Properties of a class of functions with bounded boundary rotation. Ann Polon Math 1975, 31: 311–323.MathSciNetGoogle Scholar
- Paatero V: Uber Gebiete von beschrankter Randdrehung. Ann Acad Sci Fenn Ser A 1933, 37(9):20.Google Scholar
- Noor MA, Noor KI, Al-Said EA: On certain analytic functions with bounded radius rotation. Comput Math Appl 2011, 61(10):2987–2993. 10.1016/j.camwa.2011.03.084MATHMathSciNetView ArticleGoogle Scholar
- Noor KI, Haq W, Arif M, Mustafa S: On bounded boundary and bounded radius rotations. J Inequal Appl, vol 2009., 2009: Art. ID 813687, 12Google Scholar
- Noor KI, Noor MA, Al-Said EA: On multivalent functions of bounded radius rotations. Appl Math Lett 2011, 24(7):1155–1159. 10.1016/j.aml.2011.01.042MATHMathSciNetView ArticleGoogle Scholar
- Noor KI, Noor MA, Al-Said EA: On analytic functions of bounded boundary rotation of complex order. Comput Math Appl 2011, 62: 2112–2125. 10.1016/j.camwa.2011.06.059MATHMathSciNetView ArticleGoogle Scholar
- Arif M, Noor KI, Raza M: On a class of analytic functions related with generalized Bazilevic type functions. Comput Math Appl 2011, 61: 2456–2462. 10.1016/j.camwa.2011.02.026MATHMathSciNetView ArticleGoogle Scholar
- Arif M, Raza M, Noor KI, Malik SN: On strongly Bazilevic functions associated with generalized Robertson functions. Math Comput Model 2011, 54: 1608–1612. 10.1016/j.mcm.2011.04.033MATHMathSciNetView ArticleGoogle Scholar
- Noor KI: On certain analytic functions related with strongly close-to-convex functions. Appl Math Comput 2008, 197: 149–157. 10.1016/j.amc.2007.07.039MATHMathSciNetView ArticleGoogle Scholar
- Noor KI: On the Hankel determinant of close-to-convex univalent functions. Int J Math Math Sci 1980, 3: 447–481.Google Scholar
- Noor KI: On the Hankel determinant problem for strongly close-to-convex functions. J Nat Geom 1997, 11: 29–34.MATHGoogle Scholar
- Noor KI: Some classes of analytic functions related with Bazilevic functions. Tamkang J Math 1997, 28(3):201–204.MATHMathSciNetGoogle Scholar
- Noor KI, Al-Bany SA: On Bazilevic functions. Int J Math Math Sci 1987, 10(1):79–88. 10.1155/S0161171287000103MATHMathSciNetView ArticleGoogle Scholar
- Pommerenke C: On close-to-convex analytic functions. Trans Am Math Soc 1965, 114: 176–186. 10.1090/S0002-9947-1965-0174720-4MATHMathSciNetView ArticleGoogle Scholar
- Thomas DK: On Bazilevic functions. Trans Am Math Soc 1968, 132: 353–361.Google Scholar
- Noonan JW, Thomas DK: On the Hankel determinant of areally mean p-valent functions. Proc Lond Math Soc 1972, 25(3):503–524. 10.1112/plms/s3-25.3.503MATHMathSciNetView ArticleGoogle Scholar
- Dienes P: The Taylor series. Dover, New York; 1957.Google Scholar
- Edrei A: Sur les determinants recurrents et less singularities d'une fonction donee por son developpement de Taylor. Compos Math 1940, 7: 20–88.MathSciNetGoogle Scholar
- Polya G, Schoenberg IJ: Remarks on de la Vallee Poussin means and convex conformal maps of the circle. Pac J Math 1958, 8: 259–334.MathSciNetView ArticleGoogle Scholar
- Cantor DG: Power series with integral coefficients. Bull Am Math Soc 1963, 69: 362–366. 10.1090/S0002-9904-1963-10923-4MATHMathSciNetView ArticleGoogle Scholar
- Noor KI: Hankel determinant problem for functions of bounded boundary rotations. Rev Roum Math Pures Appl 1983, 28: 731–739.MATHGoogle Scholar
- Pommerenke C: On starlike and close-to-convex analytic functions. Proc Lond Math Soc 1963, 13: 290–304. 10.1112/plms/s3-13.1.290MATHMathSciNetView ArticleGoogle Scholar
- Noor KI, Al-Naggar IA: Hankel determinant problem. J Nat Geom 1998, 14(2):133–140.MATHMathSciNetGoogle Scholar
- Golusin GM: On distortion theorem and coefficients of univalent functions. Math Sb 1946, 19: 183–203.MathSciNetGoogle Scholar
- Polatoglu Y, Bolcal M: Some results on the Janowski's starlike functions of complex order. arXiv:math/0007133v1Google 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.