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A general solution of the Fekete-Szegö problem
Boundary Value Problems volume 2013, Article number: 98 (2013)
Abstract
In the paper we introduce general classes of analytic functions defined by the Hadamard product. The Fekete-Szegö problem is completely solved in these classes of functions. Some consequences of the main results for new or well-known classes of functions are also pointed out.
MSC:30C45, 30C50, 30C55.
1 Introduction
Let denote the class of functions which are analytic in and let denote the class of functions normalized by . Each function can be expressed as
By , we denote the class of functions , which are univalent in .
A typical problem in geometric function theory is to study a functional made up of combinations of the coefficients of the original function. Usually, there is a parameter over which the extremal value of the functional is needed. The paper deals with one important functional of this type: the Fekete-Szegö functional. The classical Fekete-Szegö functional is defined by
and it is derived from the Fekete-Szegö inequality. The problem of maximizing the absolute value of the functional in subclasses of normalized functions is called the Fekete-Szegö problem. The mathematicians who introduced the functional, M. Fekete and G. Szegö [1], were able to bound the classical functional in the class by . Later Pfluger [2] used Jenkin’s method to show that this result holds for complex μ such that . Keogh and Merkes [3] obtained the solution of the Fekete-Szegö problem for the class of close-to-convex functions. Ma and Minda [4, 5] gave a complete answer to the Fekete-Szegö problem for the classes of strongly close-to-convex functions and strongly starlike functions. In the literature, there exists a large number of results about inequalities for corresponding to various subclasses of (see, for instance, [1–23]).
In the paper, we consider the classes of functions which generalize these subclasses of functions.
We say that a function is subordinate to a function , and write (or simply ), if and only if there exists a function
such that . In particular, if G is univalent in we have the following equivalence:
For functions of the forms
by we denote the Hadamard product (or convolution) of f and g, defined by
Let α be complex parameter and let , , be of the form
By we denote the class of functions such that
and
Moreover, let us put
It is clear that the class contains functions such that
We denote by the class of functions for which there exist a function such that () and
Moreover, let us denote .
In particular, the classes
are the well-known classes of α-convex Mocanu functions [24], starlike functions and convex functions, respectively. The class is the well-known class of close-to convex functions with argument .
The object of the paper is to solve the Fekete-Szegö problem in the defined classes of functions. Moreover, we find sharp bounds for the second and third coefficient in these classes. Some remarks depicting consequences of the main results are also mentioned.
2 The main results
The following lemmas will be required in our present investigation.
Lemma 1 [3]
If , (), then
The result is sharp. The functions
are the extremal functions.
Theorem 1 Let
If , then
where
The results are sharp.
Proof Let . Then there exists a function , (), such that
It is easy to verify that
where
Thus, by (7), we have
which by Lemma 1 gives sharp estimation (2). Let μ be a complex number. Then, by (10) and (11) we obtain
where γ is defined by (6). Thus, by Lemma 1, we have (4). Let functions satisfy the conditions
Then the functions belong to the class and they realize the equality in the estimation (4). Thus, the results are sharp. Putting in (4) we get the sharp estimation (3). □
Theorem 2 Let . If , then
where
The results are sharp.
Proof Let , where
Since
the results follow from Theorem 1. □
If we put in Theorem 1, then we obtain the following theorem.
Theorem 3 Let (). If , then
where
The results are sharp.
Theorem 4 Let . If , then
where
The results (12) and (13) are sharp for and , respectively.
Proof Let . Then there exists a function and functions ,
such that
Thus, by (8), we have
and by Lemma 1, we obtain the sharp estimation (14). Let μ be a complex number. Then, by (18), (19) and Lemma 1 we have
or equivalently
where , , , D are defined by (15) and (16). Thus, we obtain
and, in consequence, by Lemma 1 we have (12). It is easy to verify that the equality in (22) is attained by choosing , if , or , if , . Therefore, we consider functions such that
and
respectively. Then the functions belong to the class and they realize the equality in the estimation (12) for . Putting in (12), we get the sharp estimation (13). □
The following theorem gives the complete sharp estimation of the Fekete-Szegö functional in the class .
Theorem 5 Let . If , then
where , , , D are defined by (15) and (16). The result is sharp.
Proof From Theorem 4, we have sharp estimation (23) for . Let now and . Then, by (20) and Lemma 1 we have
where
Simply calculations give that the function v attains a maximum in the interval at the point . Thus, we have (23) for and . Moreover, the equality in (20) is attained by choosing the functions , , for , i.e. , and , . Therefore, the result is sharp for and .
Next, let and . Then, by (20) and Lemma 1 we have
where
Since the function attains a maximum in the interval at the point , we have the estimation (23) for and . The equality in (20) is attained by choosing the functions , , for , i.e. , and , .
Finally, let us assume . Then, by (20) we have
where
Since F is the continuous function on , by (24) we have
where K is the set of critical points of the function F in T. It is easy to verify that
If , then
Moreover, we have
Thus, we obtain
which by (25) gives (23) for . The equality in (24) is attained by choosing and . Therefore, the result is sharp and the proof is completed. □
Putting in Theorem 5 we obtain the following theorem.
Theorem 6 Let . If , then
where , , , D are defined by (15) and (16). The result is sharp.
3 Applications
If we put and in Theorem 2, then we obtain the following two corollaries.
Corollary 1 Let . If , then
where
The results are sharp.
Corollary 2 Let . If , then
where
The results are sharp.
Choosing the function p in Theorems 1-6, we can obtain several new results.
Let a, b be complex number, , , and let
It is clear, that
Thus, by Theorems 1-3 and 5, we obtain the following four corollaries.
Corollary 3 Let (). If , then
where
The results are sharp.
Corollary 4 Let . If , then
where
The results are sharp.
Corollary 5 Let (). If , then
where
The results are sharp.
Corollary 6 Let . If , then
where
The result is sharp.
Let and let
It is easy to verify, that
Thus, by Theorems 1-3 and 5, we obtain the following four corollaries.
Corollary 7 Let (). If , then
where
The results are sharp.
Corollary 8 Let . If , then
where
The results are sharp.
Corollary 9 Let (). If , then
where
The results are sharp.
Corollary 10 Let . If , then
where
The result is sharp.
Let , . Note that is the convex domain contained in the right half plane, with . More precisely, it is the elliptic domain for , the hyperbolic domain for and the parabolic domain for .
Let us denote by the univalent function, which maps the unit disc onto the conic domain with . Obviously, the function is convex in . It is easy to check that if and only if
The following lemma gives coefficients estimates for the function.
Lemma 2 [13]
Let (). Then
where and is the complete elliptic integral of first kind.
Using Lemma 1 in Theorems 1-5 we obtain the solutions of the Fekete-Szegö problem for the classes , , , .
Remark 1 The classes , , reduced to well-known subclasses by judicious choices of the parameters; see, for example [1–28]. In particular, they generalize several well-known classes defined by linear operators, which were investigated in earlier works. Also, the obtained results generalize several results obtained in these classes of functions.
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
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An erratum to this article is available at http://dx.doi.org/10.1186/1687-2770-2014-50.
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Dziok, J. A general solution of the Fekete-Szegö problem. Bound Value Probl 2013, 98 (2013). https://doi.org/10.1186/1687-2770-2013-98
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DOI: https://doi.org/10.1186/1687-2770-2013-98