Abstract
In the paper we introduce general classes of analytic functions defined by the Hadamard product. The FeketeSzegö problem is completely solved in these classes of functions. Some consequences of the main results for new or wellknown classes of functions are also pointed out.
MSC: 30C45, 30C50, 30C55.
Keywords:
analytic functions; FeketeSzegö problem; subordination; Hadamard product1 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 FeketeSzegö functional. The classical FeketeSzegö functional is defined by
and it is derived from the FeketeSzegö inequality. The problem of maximizing the absolute value of the functional in subclasses of normalized functions is called the FeketeSzegö 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 FeketeSzegö problem for the class of closetoconvex functions. Ma and Minda [4,5] gave a complete answer to the FeketeSzegö problem for the classes of strongly closetoconvex 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, [123]).
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:
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
In particular, the classes
are the wellknown classes of αconvex Mocanu functions [24], starlike functions and convex functions, respectively. The class is the wellknown class of closeto convex functions with argument .
The object of the paper is to solve the FeketeSzegö 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]
The result is sharp. The functions
are the extremal functions.
Theorem 1Let
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). □
where
The results are sharp.
Since
the results follow from Theorem 1. □
If we put in Theorem 1, then we obtain the following theorem.
where
The results are sharp.
where
The results (12) and (13) are sharp forand, 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 FeketeSzegö functional in the class .
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
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
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.
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.
where
The results are sharp.
where
The results are sharp.
Choosing the function p in Theorems 16, we can obtain several new results.
Let a, b be complex number, , , and let
It is clear, that
Thus, by Theorems 13 and 5, we obtain the following four corollaries.
where
The results are sharp.
where
The results are sharp.
where
The results are sharp.
where
The result is sharp.
It is easy to verify, that
Thus, by Theorems 13 and 5, we obtain the following four corollaries.
where
The results are sharp.
where
The results are sharp.
where
The results are sharp.
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]
whereandis the complete elliptic integral of first kind.
Using Lemma 1 in Theorems 15 we obtain the solutions of the FeketeSzegö problem for the classes , , , .
Remark 1 The classes , , reduced to wellknown subclasses by judicious choices of the parameters; see, for example [128]. In particular, they generalize several wellknown 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.
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