Research

# A general solution of the Fekete-Szegö problem

Jacek Dziok

Author Affiliations

Institute of Mathematics, University of Rzeszów, Rzeszów, 35-310, Poland

Boundary Value Problems 2013, 2013:98  doi:10.1186/1687-2770-2013-98

 Received: 24 January 2013 Accepted: 9 April 2013 Published: 22 April 2013

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.

### 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.

##### Keywords:
analytic functions; Fekete-Szegö problem; subordination; Hadamard product

### 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

(1)

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 1Let

If, then

(2)

(3)

(4)

where

(5)

(6)

The results are sharp.

Proof Let . Then there exists a function , (), such that

(7)

It is easy to verify that

(8)

(9)

where

Thus, by (7), we have

(10)

(11)

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 2Let. 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 3Let (). If, then

where

The results are sharp.

Theorem 4Let. If, then

(12)

(13)

(14)

where

(15)

(16)

The results (12) and (13) are sharp forand, respectively.

Proof Let . Then there exists a function and functions ,

such that

(17)

Thus, by (8), we have

(18)

(19)

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

(20)

or equivalently

(21)

where , , , D are defined by (15) and (16). Thus, we obtain

(22)

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 5Let. If, then

(23)

where, , , Dare 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

(24)

where

Since F is the continuous function on , by (24) we have

(25)

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 6Let. If, then

where, , , Dare 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 1Let. If, then

where

The results are sharp.

Corollary 2Let. 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 3Let (). If, then

where

The results are sharp.

Corollary 4Let. If, then

where

The results are sharp.

Corollary 5Let (). If, then

where

The results are sharp.

Corollary 6Let. 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 7Let (). If, then

where

The results are sharp.

Corollary 8Let. If, then

where

The results are sharp.

Corollary 9Let (). If, then

where

The results are sharp.

Corollary 10Let. 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

whereandis 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.

### Competing interests

The author declares that they have no competing interests.

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

### References

1. Fekete, M, Szegö, G: Eine Bemerkung über ungerade schlichte Funktionen. J. Lond. Math. Soc.. 8, 85–89 (1933)

2. Pfluger, A: The Fekete-Szegö inequality for complex parameters. Complex Var. Theory Appl.. 7, 149–160 (1986). Publisher Full Text

3. Keogh, FR, Merkes, EP: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc.. 20, 8–12 (1969). Publisher Full Text

4. Ma, W, Minda, D: An internal geometric characterization of strongly starlike functions. Ann. Univ. Mariae Curie-Skl̄odowska, Sect. A. 45, 89–97 (1991)

5. Ma, W, Minda, D: Coefficient inequalities for strongly close-to-convex functions. J. Math. Anal. Appl.. 205, 537–553 (1997). Publisher Full Text

6. Abdel-Gawad, HR, Thomas, DK: The Fekete-Szegö problem for strongly close-to-convex functions. Proc. Am. Math. Soc.. 114, 345–349 (1992)

7. Cho, NE, Owa, S: On the Fekete-Szegö and argument inequalities for strongly close-to-star functions. Math. Inequal. Appl.. 5, 697–705 (2002)

8. Choi, JH, Kim, YC, Sugawa, T: A general approach to the Fekete-Szegö problem. J. Math. Soc. Jpn.. 59(3), 707–727 (2007). Publisher Full Text

9. Bhowmik, B, Ponnusamy, S, Wirths, K-J: On the Fekete-Szegö problem for concave univalent functions. J. Math. Anal. Appl.. 373, 432–438 (2011). Publisher Full Text

10. Darus, M, Thomas, DK: On the Fekete-Szegö theorem for close-to-convex functions. Math. Jpn.. 44, 507–511 (1996)

11. Darus, M, Shanmugam, TN, Sivasubramanian, S: Fekete-Szegö inequality for a certain class of analytic functions. Mathematica. 49, 29–34 (2007)

12. Jakubowski, ZJ, Zyskowska, K: On an estimate of a functional in the class of holomorphic univalent functions. Math. Bohem.. 118(3), 281–296 (1993)

13. Kanas, S: Coefficient estimates in subclasses of the Carathéodory class related to conic domains. Acta Math. Univ. Comen.. 74, 149–161 (2005)

14. Kanas, S, Darwish, HE: Fekete-Szegö problem for starlike and convex functions of complex order. Appl. Math. Lett.. 23, 777–782 (2010). Publisher Full Text

15. Koepf, V: On the Fekete-Szegö problem for close-to-convex functions. Proc. Am. Math. Soc.. 101, 89–95 (1987)

16. Koepf, W: On the Fekete-Szegö problem for close-to-convex functions II. Arch. Math.. 49, 420–433 (1987). Publisher Full Text

17. London, RR: Fekete-Szegö inequalities for close-to-convex functions. Proc. Am. Math. Soc.. 117, 947–950 (1993)

18. Mishra, AK, Gochhayat, P: The Fekete-Szegö problem for k-uniformly convex functions and for a class defined by the Owa-Srivastava operator. J. Math. Anal. Appl.. 347, 563–572 (2008). Publisher Full Text

19. Orhan, H, Deniz, E, Răducanu, D: The Fekete-Szegö problem for subclasses of analytic functions defined by a differential operator related to conic domains. Comput. Math. Appl.. 59, 283–295 (2010). Publisher Full Text

20. Orhan, H, Răducanu, D: Fekete-Szegö problem for strongly starlike functions associated with generalized hypergeometric functions. Math. Comput. Model.. 50, 430–438 (2009). Publisher Full Text

21. Sim, YJ, Kwon, OS, Cho, NE, Srivastava, HM: Some classes of analytic functions associated with conic regions. Taiwan. J. Math.. 16, 387–408 (2012)

22. Srivastava, HM, Mishra, AK: Applications of fractional calculus to parabolic starlike and uniformly convex functions. Comput. Math. Appl.. 39, 57–69 (2000)

23. Srivastava, HM, Mishra, AK, Das, MK: The Fekete-Szegö problem for a subclass of close-to-convex functions. Complex Var. Theory Appl.. 44, 145–163 (2001). Publisher Full Text

24. Mocanu, PT: Une propriété de convexité g énéralisée dans la théorie de la représentation conforme. Mathematica (Cluj). 11(34), 127–133 (1969)

25. Aouf, MK, Dziok, J: Distortion and convolutional theorems for operators of generalized fractional calculus involving Wright function. J. Appl. Anal.. 14(2), 183–192 (2008)

26. Dziok, J: Applications of the Jack lemma. Acta Math. Hung.. 105, 93–102 (2004)

27. Dziok, J: Classes of functions defined by certain differential-integral operators. J. Comput. Appl. Math.. 105, 245–255 (1999). Publisher Full Text

28. Dziok, J: Inclusion relationships between classes of functions defined by subordination. Ann. Pol. Math.. 100, 193–202 (2011). Publisher Full Text