#### Abstract

For -valently Janowski starlike and convex functions defined by applying subordination for the generalized Janowski function, the sharp upper bounds of a functional related to the Fekete-Szegö problem are given.

#### 1. Introduction

Let denote the family of functions normalized by which are analytic in the open unit disk . Furtheremore, let be the class of functions of the form which are analytic and satisfy in . Then, a function is called the Schwarz function. If satisfies the following condition for some complex number , then is said to be -valently starlike function of complex order . We denote by the subclass of consisting of all functions which are -valently starlike functions of complex order . Similarly, we say that is a member of the class of -valently convex functions of complex order in if satisfies for some complex number .

Next, let and . Then, the condition of the definition of is equivalent to

We denote by the distance between the boundary line of the half plane satisfying the condition (1.5) and the point . A simple computation gives us that that is, that is always equal to regardless of . Thus, if we consider the circle with center at and radius , then we can know the definition of means that is covered by the half plane separated by a tangent line of and containing . For , the same things are discussed by Hayami and Owa .

Then, we introduce the following function: which has been investigated by Janowski . Therefore, the function given by (1.7) is said to be the Janowski function. Furthermore, as a generalization of the Janowski function, Kuroki et al.  have investigated the Janowski function for some complex parameters and which satisfy one of the following conditions: Here, we note that the Janowski function generalized by the conditions (1.8) is analytic and univalent in , and satisfies . Moreover, Kuroki and Owa  discussed the fact that the condition can be omitted from among the conditions in (1.8)-(i) as the conditions for and to satisfy . In the present paper, we consider the more general Janowski function as follows: for some complex parameter and some real parameter . Then, we don't need to discuss the other cases because for the function: letting and replacing by in (1.10), we see that maps onto the same circular domain as .

Remark 1.1. For the case in (1.9), we know that maps onto the following half plane: and for the case in (1.9), maps onto the circular domain

Let and be analytic in . Then, we say that the function is subordinate to in , written by if there exists a function such that . In particular, if is univalent in , then if and only if

We next define the subclasses of by applying the subordination as follows: where , . We immediately know that

Then, we have the next theorem.

Theorem 1.2. If , then , where Espesially, if and only if where .

Proof. Supposing that , it follows from Remark 1.1 that that is, that This means that which implies that Therefore, , where . The converse is also completed.
Next, for the case , by the definition of the class , if a tangent line of the circle containing the point is parallel to the straight line , and the image by is covered by the circle , then there exists a non-zero complex number with and such that , where is the distance between the tangent line and the point . Now, for the function , the image is equivalent to and the point on can be written by Further, the tangent line of the circle through each point is parallel to the straight line . Namely, can be represented by which implies that Then, we see that the distance between the point and the above tangent line of the circle is Therefore, if the subordination holds true, then where Finally, setting , the proof of the theorem is completed.

Noonan and Thomas [8, 9] have stated the th Hankel determinant as This determinant is discussed by several authors with . For example, we can know that the functional is known as the Fekete-Szegö problem, and they consider the further generalized functional , where and is some real number (see, ). The purpose of this investigation is to find the sharp upper bounds of the functional for functions or .

#### 2. Preliminary Results

We need some lemmas to establish our results. Applying the Schwarz lemma or subordination principle.

Lemma 2.1. If a function , then Equality is attained for for any .

The following lemma is obtained by applying the Schwarz-Pick lemma (see, e.g., ).

Lemma 2.2. For any functions , holds true. Namely, this gives us the following representation: for some .

#### 3. -Valently Janowski Starlike Functions

Our first main result is contained in

Theorem 3.1. If , then with equality for

Proof. Let . Then, there exists the function such that which means that where . Thus, by the help of the relation in Lemma 2.2, we see that Then, by Lemma 2.1, supposing that without loss of generality, and applying the triangle inequality, it follows that

Especially, taking in Theorem 3.1, we obtain the following corollary.

Corollary 3.2. If , then with equality for

Furthermore, putting and for some in Theorem 3.1, we arrive at the following result by Hayami and Owa [2, Theorem  3].

Corollary 3.3. If , then with equality for

#### 4. -Valently Janowski Convex Functions

Similarly, we consider the functional for -valently Janowski convex functions.

Theorem 4.1. If , then with equality for where represents the ordinary hypergeometric function and represents the confluent hypergeometric function.

Proof. By the help of the relation (1.17) and Theorem 3.1, if , then where is one of the values in Theorem 3.1. Then, dividing the both sides by and replacing by , we obtain the theorem.

Now, letting in Theorem 4.1, we have the following corollary.

Corollary 4.2. If , then wiht equality for where represents the ordinary hypergeometric function and represents the confluent hypergeometric function.

Moreover, we suppose that and for some . Then, we arrive at the result by Hayami and Owa [2, Theorem  4].

Corollary 4.3. If , then with equality for where represents the ordinary hypergeometric function.