- Open Access
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.
- Robertson function
- strongly Bazilevic functions
- bounded boundary rotations
- Hankel determinant
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
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.
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