International Journal of Mathematics and Mathematical Sciences

Volume 2014 (2014), Article ID 570361, 10 pages

http://dx.doi.org/10.1155/2014/570361

Research Article

On a Subclass of Analytic Functions Related to a Hyperbola

Jagannath Patel¹ and Ashok Kumar Sahoo²

¹Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar 751004, India

²Department of Mathematics, Institute of Technical Education and Research, Jagmohan Nagar, Khandagiri, Bhubaneswar 751030, India

Received 14 January 2014; Accepted 12 April 2014; Published 7 May 2014

Academic Editor: Gelu Popescu

Copyright © 2014 Jagannath Patel and Ashok Kumar Sahoo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The object of the present investigation is to solve Fekete-Szegö problem and determine the sharp upper bound to the second Hankel determinant for a new class of analytic functions in the unit disk. We also obtain a sufficient condition for an analytic function to be in this class.

1. Introduction and Preliminaries

Let be the class of functions of the form:

which are analytic in the open unit disk .

A function is said to be starlike function of order and convex function of order , respectively, if and only if and , for and for all . By usual notations, we denote these classes of functions by and , respectively. We write and , the familiar subclasses of starlike functions and convex functions in .

Furthermore, a function is said to in the class , if it satisfies the inequality:

Note that is a subclass of close-to-convex functions of order in .

Let denote the class of analytic functions of the form:

satisfying the condition in .

Let the functions and be analytic in . We say that is subordinate to , written as or , if there exists a Schwarz function , which (by definition) is analytic in with and . Furthermore, if the function is univalent in , then we have the following equivalence relation (cf., e.g., [1]):

For the functions analytic in and given by the power series

their Hadamard product (or convolution), denoted by is defined as

Note that is analytic in .

The Gauss hypergeometric function is defined by the infinite series

where denotes the Pochhammer symbol (or shifted factorial) given, in terms of the Gamma function , by

We note that the series, given by (7), converges absolutely for and hence the function represents an analytic function in the unit disc [2].

We further observe that the Gauss hypergeometric function plays an important role in the study of various properties and characteristics of subclasses of univalent/multivalent functions in geometric function theory (cf., e.g. [3–5]). In our present investigation, we consider the incomplete beta function , defined by

By making use of the Hadamard product and the function , Carlson and Shaffer [6] defined the linear operator by

If is given by (1), then it follows from (10) that

The operator extends several operators introduced and studied by earlier researchers in geometric function theory.  For example, , the well-known Ruscheweyh derivative operator [7] of and , the familiar Owa-Srivastava fractional differential operator [8] of .

With the aid of the linear operator , we introduce a subclass of as follows.

Definition 1. A function is said to be in the class , if it satisfies the following subordination relation:

where the power in the right hand side of (13) indicates the principal branch. Note that if , then by (13)

We denote by , the class of functions satisfying the subordination condition:

In fact, by suitably specializing the parameters , and in the class , we can obtain several subclasses of .

Remark 2. To bring out the geometrical significance of the class , we set

and note that

which gives or . Letting , we deduce that

which on simplification reduces to . Thus, is the interior of the right half branch of the hyperbola . Hence, if , then the set of values for lie in , where is given by (16).

Fekete and Szegö [9] defined the Hankel determinant of a function , given by (1) as

In our present investigation, we also consider the second Hankel determinant of , given by

It is known [10] that if given by (1) is analytic and univalent in , then the sharp inequality holds. For a family of functions in of the form (1), the more general problem of finding the sharp upper bounds for the functionals is popularly known as Fekete-Szegö problem for the class . The Fekete-Szegö problem for the known classes of univalent functions, starlike functions, convex functions, and close-to-convex functions has been completely settled [9, 11–18]. Recently, Janteng et al. [19, 20] have obtained the sharp upper bounds to the second Hankel determinant for the family of functions in whose derivatives have positive real part in . For initial work on the class , one may refer to the paper by MacGregor [21].

Our objective in the present paper is to solve the Fekete-Szegö problem and also to determine the sharp upper bound to the second Hankel determinant for the class by following the techniques devised by Libera and Złotkiewicz [22, 23]. The criteria for functions in to be in this class are also obtained.

To establish our main results, we will need the following results about the functions belonging to the class .

Lemma 3. Let the function , given by (3), be a member of the class . Then

for some complex numbers satisfying and . The estimates in (21) and (22) are sharp.

We note that the estimate (21) is contained in [10]; the estimate (22) is obtained by Ma and Minda [24]; the results in (23) and (24) are due to Libera and Złotkiewicz [23] (see also [22]).

2. Main Results

Unless otherwise mentioned, we assume throughout the sequel that

Now, we determine the sharp upper bound for the functional for functions of the form (1) belonging to the class .

Theorem 4. Let and . If the function , given by (1), belongs to the class , then for any

The estimate in (26) is sharp.

Proof. Since , by (14) we have

where is given by (3). It is easily seen that

Writing the series expansion of given by (11), , in (27) and equating the coefficients of in the resulting equation, we obtain

Thus for any ,

and by using (22) in the above expression, we get

which, upon simplification, gives the required assertion of Theorem 4.

Equality in (26) holds for the function defined in by

where the function is given by (16). This completes the proof of Theorem 4.

Theorem 5. Let and . If the function , given by (1), belongs to the class , then

The estimates are sharp.

Proof. First, we assume that . Then

so that by (26), we obtain

Next, let

Then, a routine calculation yields

and by using (26) again, we get

Finally, if , then

Thus, by (26), we have

The estimates are sharp for the function defined in by

where the function is given by (16) and the proof of Theorem 5 is completed.

Using (21) in (29) and putting and , respectively, in Theorem 5, we get the following.

Corollary 6. Let . If the function , given by (1), belongs to the class , then

The estimates in (44) and (46) are sharp for the function defined by

whereas the estimate in (45) is sharp for the function given by

where the function is given by (16).

Letting and in Theorem 8, we obtain the following.

Corollary 7. If the function , given by (1), belongs to the class , then

The estimates are sharp for the function defined in by

where is given by (16).

Next, we find the sharp upper bound for the fourth coefficient of functions in the class .

Theorem 8. Let the function , given by (1), belong to the class . Then

and the estimate in (51) is sharp.

Proof. From (31), we have

Since the functions and are in the class simultaneously, we assume without loss of generality that . For convenience of notation, we write . Now, by using (23) and (24) in (52), we deduce that

for some complex numbers and .

Applying the triangle inequality in the above expression followed by the replacement of with in the resulting equation, we obtain

We next maximize the function on the closed rectangle . Since

we have for and . Thus, cannot have a maximum in the interior on the closed rectangle . Therefore, for fixed

where

A routine calculation yields

for or . Since and , we conclude that the maximum of is attained at . Thus, the upper bound of the function corresponds to . Putting in (54), we get our desired estimate (51).

Equality in (51) holds for the function defined by

where is given by (16).

In the following theorem, we find the sharp upper bound to the second Hankel determinant for the class .

Theorem 9. Let and . If the function , given by (1), belongs to the class , then

The estimate in (60) is sharp.

Proof. From (29), (30), and (31), we deduce that

As in Theorem 8, we assume without loss of generality that and for convenience of notation, we write . By using (23) and (24) in (61), we get

Now, by applying the triangle inequality in (62) and replacing by in the resulting equation, we get

We next maximize the function on the closed rectangle . Since

for and , it follows that cannot have a maximum in the interior on the closed rectangle . Thus, for fixed

where

. and . Differentiating with respect to , we deduce that

for or

Since and

by the hypothesis, we conclude that the maximum value of is attained at so that the upper bound of the function corresponds to and . Thus, by letting and in (63), we get the estimate (60).

The estimate in (60) is sharp for the function given by (48). This completes the proof of Theorem 9.

Putting and in Theorem 9, we get the following.

Corollary 10. If the function , given by (1) belongs to the class , then

and the estimate is sharp for the function defined by

where the function is given by (16).

Theorem 11. Let and . If satisfies the following inequality

then

The result is the best possible.

Proof. We define the function by

Choosing the principal branch in the right hand side in (74), we note that is analytic in with . Furthermore, logarithmically differentiating (74) and using the identity (12) in the resulting equation, we find that

We claim that for all . If not, then there exists a point such that

and let . Now, by applying Jack's lemma [25], we have

From (75) and (77), we obtain

which contradicts the hypothesis (72). Thus, we conclude that for all and (74) yields the required subordination relation (73).

To see that the result is the best possible, we consider the function defined by

from which it follows that

Thus, satisfies the subordination relation (73). On differentiating the expression in (80) followed by the use of the identity (12) in the resulting equation, we deduce that

This implies that

and the proof of Theorem 11 is completed.

In the special case , we get the following sufficient condition for the class .

Corollary 12. Let and . If satisfies the following inequality:

then . The result is the best possible for the function given by (47).

Letting and in Theorem 11, we obtain the following.

Corollary 13. If and satisfies

then . The result is the best possible for the function defined by

where the function is given by (16).

Theorem 14. Let and . If satisfies the following subordination relation:

then

where

The bound in (88) is the best possible.

Proof. From (86), we get

where we choose the principal branch in (89). Taking logarithmic differentiation in (89) and using the identity (12) in the resulting equation, we deduce that

Using the following well-known estimates [21]

in (90), we get

which is certainly positive for , where is given by (88).

To show that the result is the best possible, we consider the function defined by

Noting that

for , we conclude that the bound is the best possible. This proves Theorem 14.

Taking in Theorem 14, we get the following.

Corollary 15. If and , then

where

The bound is the best possible for the function , given by (47).

Setting and in Theorem 14, we get the following.

Corollary 16. If satisfies

then

where

The bound is the best possible for the function , given in Corollary 13.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the reviewers for their constructive suggestions and comments which improved the presentation of the paper.

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