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A certain class of analytic functions and the growth rate of Hankel determinant
Journal of Inequalities and Applications volume 2012, Article number: 309 (2012)
The growth rate of coefficients and Hankel determinant for any class of analytic functions is well known. In this present investigation, we define a class of analytic functions related to strongly close-to-convex functions. We investigate different interesting properties for this class including arc length, the growth rate of coefficients and the growth rate of Hankel determinant by using the method of Noonan and Thomas. Several well-known results appear as special cases from our results.
We denote by A the class of functions f which are analytic in the open unit disc and of the form
Also let us denote by the class of functions h with , which are analytic in E and satisfying
where , and . This class has been investigated by Padmanabhan and Parvatham . For , we obtain the class , introduced by Pinchuk  and for , , we obtain the class P of functions with a positive real part.
The class was first introduced and investigated by the second author  as a generalization of close-to-convexity. She studied its geometrical interpretation and various other interesting properties including the growth rate of coefficient differences and a radius of convexity problem. Recently, she  studied the class of analytic functions corresponding to strongly close-to-convex functions. She employed a modification to a method of Pommerenke  to investigate the growth rate of Hankel determinant problems regarding this class.
In this paper, we define a class of analytic functions related to strongly close-to-convex functions. We investigate different interesting properties including inclusion relations, arc length, the growth rate of coefficients and the growth rate of Hankel determinant by using a different method from that given in .
We now define the following classes of analytic functions.
Definition 1.1 Let be locally univalent in E. Then, for (complex), , if and only if
We note that for , we have the class of bounded boundary rotations of order ρ introduced by Padmanabhan and Parvatham . Also, , the class of functions of bounded boundary rotations and denotes the class of convex functions of complex order.
Definition 1.2 Let . Then if and only if, for , , there exists a function such that
For , we have the class which was recently introduced and studied by Noor . For , , , is the class of strongly close-to-convex functions. Also, is the class of convex functions of order ρ. For , , , the class of reduces to the class introduced by Noor .
We need the following results in our investigation.
Lemma 1.1 A function if and only if
there exist two normalized starlike functions and such that(1.2)
The above lemma is a special case of the result discussed in .
Lemma 1.2 
Let with . Then
Lemma 1.3 
Let f be univalent and . Then there exists a number with , such that for all z, , we have
2 Some properties of the class
Following essentially the same method as given in , we can easily obtain the following result.
Theorem 2.1 The function if and only if
where and are strongly close-to-convex functions of order β.
Theorem 2.2 Let in E. Then for , where
This result is sharp.
Proof We can write
Using Lemma 1.1, we have
where and are starlike functions. Logarithmic differentiation of (2.2) gives us
which implies that
Now using distortion results for the class P, we have
The right-hand side of (2.3) is positive for , where is given by (2.1). The sharpness can be viewed from the function , given by
We note the following.
For , we have the radius of convexity for the class studied by Noor .
For , , , we have the radius of convexity for the class , proved by Noor .
For , , , , we have the radius of convexity for close-to-convex functions which is well known.
We now discuss the arc length problem and the growth rate of coefficients for the class .
Theorem 2.3 Let , for , , and . Then
where is a constant depending only on k, η, ρ, β. The exponent is sharp.
Proof We have
Using Definition 1.1, Lemma 1.1(iii) and the distortion theorem for starlike functions, we have
Using Holder’s inequality with , such that , we obtain
Since , therefore using subordination for starlike functions and Lemma 1.2, we have
The function defined by
shows that the exponent is sharp. □
By assigning different values to the parameters involved in the above theorem, we have the following interesting results.
Corollary 2.1 Let . Then
Corollary 2.2 Let . Then
Coefficient growth problems The problem of growth rate and asymptotic behavior of coefficients is well known. In the upcoming results, we investigate these problems for a different set of classes by varying different parameters.
Theorem 2.4 Let and be of the form (1.1). Then, for , , , , , we have
where is a constant depending only on k, η, ρ, β. The exponent is sharp.
Proof With , Cauchy’s theorem gives us
Using Theorem 2.3 and putting , we obtain the required result. The sharpness follows from the function defined by the relation (2.5). □
Corollary 2.3 Let and be of the form (1.1). Then, for , , we have
For , in the above corollary, we have the growth rate of coefficients problem for functions in this class and for , , gives us the growth rate of coefficient estimates for close-to-convex functions, which is well known.
3 Hankel determinant problem
The Hankel determinant of a function f of the form (1.1) is defined by
The growth rate of Hankel determinant as , when f is a member of any class of analytic functions, is well known. Pommerenke  proved that when f is an areally mean p-valent function, then for ,
and and where depends on p, q, and the function f. In particular, this shows that as for large q relative to p. In fact, for ,
The exponent is exact.
Noonan and Thomas  gave the exact growth rate of for large p relative to q, and they proved that
where depends upon p, q only and the exponent is best possible. Also, for univalent functions, Pommerenke  has proved that for ,
where , which in particular shows that
Pommerenke  has shown that if f is starlike, then for ,
Also, for , it is shown  that for , ,
where depends upon p, q and f only. The exponent is best possible.
Following the notation of Noonan and Thomas , we define the following.
Definition 3.1 Let be a non-zero complex number. Then for , given by (1.1), we define
The following two lemmas are due to Noonan and Thomas  which are essential in our investigations.
Lemma 3.1 Let and let the Hankel determinant of f be defined by (3.1). Then, writing , we have
Lemma 3.2 With and any integer,
We also need the following remark given in .
Remark 3.1 Consider any determinant of the form
with , and , . Thus,
where is the symmetric group on q elements, and is either +1 or −1. Thus, in the expansion of D, each summand has q factors, and the sum of the subscripts of the factors of each summand is .
Now, let n be given and be as in Lemma 3.1, then each summand in the expansion of is of the form
We now prove the following.
Theorem 3.1 Let and let the Hankel determinant of f, for , , be defined by (3.1). Then, for and , we have
where depends only on k, η, ρ, β and q.
Proof Since , there exists such that
Now, for , any non-zero complex number and , we consider for ,
where we have used Lemma 1.1(iii). Using Lemma 1.3, we have
By employing distortion results for starlike functions and simplifying, we obtain, from (3.3),
Using Holder’s inequality, with , , such that , we can write
Proceeding in a similar way as in Theorem 2.3, we have
Subordination for starlike functions further yields
where depends only on k, η, β and j.
Now, applying Lemma 3.2 and putting (), we have for , ,
We now estimate the growth rate of . For , and from Theorem 2.4, it follows that
For , we use Remark 3.1 together with Lemma 3.1, to have
where depends only on k, η, ρ, β and q. □
By giving special values to the parameters involved in the above theorem, we obtain the following interesting results.
Corollary 3.1 Let and be defined as in (1.1). Then, for , ,
where depends only on k, ρ, β and q.
Noor  studied the above corollary with a different method.
Corollary 3.2 Let and be defined as in (1.1). Then, for , ,
where depends only on k and q.
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The authors would like to thank the reviewers of this article for their valuable comments on the earlier version of this paper.
The authors declare that they have no competing interests.
WH carried out all the calculations and drafted the manuscript. KIN provided results and ideas which were used in proofs of main theorems.
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Ul-Haq, W., Noor, K.I. A certain class of analytic functions and the growth rate of Hankel determinant. J Inequal Appl 2012, 309 (2012). https://doi.org/10.1186/1029-242X-2012-309
- analytic functions
- Hankel determinant