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Upper Bound of Partial Sums Determined by Matrix Theory

Rabha W. Ibrahim

Institute of Mathematical Sciences, University Malaya, Malaysia

Abstract: One of the major problems in the geometric function theory is the coefficients bound for functional and partial sums. The important method, for this purpose, is the Hankel matrix. Our aim is to introduce a new method to determine the coefficients bound, based on the matrix theory. We utilize various kinds of matrices, such as Hilbert, Hurwitz and Turan. We illustrate new classes of analytic function in the unit disk, depending on the coefficients of a particular type of partial sums. This method shows the effectiveness of the new classes. Our results are applied to the well known classes such as starlike and convex. One can illustrate the same method on other classes.

Keywords: analytic function, univalent function, unit disk, partial sums, coefficients bound

Citation: Rabha W. Ibrahim, Upper Bound of Partial Sums Determined by Matrix Theory. Turkish Journal of Analysis and Number Theory, 3 (6). 149-153. January 2015. doi:10.12691/tjant-3-6-2

1. Introduction

The Hankel determinant represents a major part in the theory of singularities [1,2]. In addition, it utilizes in the investigation of power series with integral coefficients [3]. Also, it appears in the study of meromorphic functions [4], and various properties of these determinants can be found in [5]. It is well known that the Fekete-Szego functional This functional is further generalized as for some (real or complex). Fekete and Szego introduced sharp bounds of for real of univalent functions. It is a very important combination of the two coefficients which describes the area problems posted earlier by Gronwall. Furthermore, researchers considered the functional (see [6]). Babalola [7] determined the Hankel determinant for some subclasses of analytic functions. Ibrahim [8] computed the Hankel determinant for fractional differential operator in the open unit disk.

Partial sums are studied widely in the univalent function theory. Szeg [9] proved that if the function is starlike, then its partial sums are starlike for . Moreover, if is convex, then its partial sums is convex for Later Owa [10] imposed the starlikeness and convexity for special case of In addition, Darus and Ibrahim [11] specified the assumptions, which indicated that the partial sums of functions of bounded turning are also of bounded turning. Recently, Darus and Ibrahim [12] considered the Cesáro partial sums, it has been shown that this type of partial sums preserves the properties of the analytic functions in the open unit disk.

In this work, we deal with the partial sums of the form We introduce some classes of analytic functions defined by its partial sums. The stability of these classes is studied by utilizing Hurwitz matrices convoluting the with Hilbert matrix (a special type of Hankel matrix). Moreover, we discuss some partial sums formulated under Turan determinant. The upper bound as well as the lower bound of the coefficients . This new process includes some well known results. Our outcomes depend on computational results of different order of the Turan determinant. We show that some geometric properties, of the new classes are established by computing the Turan determinant such as starlikeness and convexity.

2. Processing

Let be the class of analytic functions in and normalized by the conditions For a partial sum of the form

convoluted with the Hilbert matrix elements in the fit order, we obtain the partial sums

For the above partial sums , we let

[picture] | (1)

The minors of Hurwitz matrix for (1), are defined by

Definition 2.1 For the polynomial is called stable, asymptotically stable and unstable if and only if for all respectively.

From (1), we define the partial sums

We proceed to construct new classes based on . A computation implies

[picture] | (2)

Thus for we have the following classes:

[picture] | (3)

We call the above classes the coefficient -starlike and they denoted by . Similarly, we define the coefficient -convex, which denoted by , as follows:

[picture] | (4)

Thus for we have the following classes:

[picture] | (5)

In the same manner of the above classes, one can construct -class such as close to convex, uniformly classes and concave. Based on these classes, we can study the stability of starlikeness as well as convexity. Moreover, relations concerning these classes can be formulated such as , .

3. Outcomes

We have the following stability results for the classes and :

Theorem 3.1 Consider Then a polynomial of degree 2 is starlike stable, while of degree 3 is not stable.

Proof. By employing , in Eq. (3), polynomials of degree 2 and 3 can be expressed respectively as follows:

and

Let , thus we obtain

and

Theorem 3.2 Consider Then a polynomial of degree 2 is convex stable, while of degree 3 is not stable.

Proof. Consider Then polynomials of degree 2 and 3 can be formulated respectively as follows:

and

Let , thus we obtain

and

Consider . We deal with polynomial sequences (partial sums) satisfying the recurrent relation

[picture] | (6)

and

Define the Turan determinant as follows:

[picture] | (7)

We shall prove inequality of the form

[picture] | (8)

Theorem 3.3 Assume satisfies (6). Then

Proof. By (6), we have

this yields that

Consequently, we obtain

[picture] | (9)

By the definition of , we conclude that

[picture] | (10)

then summing (9) and (10), we arrive at the desired assertion. This completes the proof.

Theorem 3.4 Let be increasing sequence. If then

Proof. It suffices to show that By the proof of Theorem 3.3 and the fact that and we conclude that

Therefore, by the assumptions of the theorem, we have Hence by induction we obtain

Define a function then satisfies the following property :

Proposition 3.1 For we have

Proof. A calculation implies that

Theorem 3.5 For we have

where

Proof. We observe that

[picture] | (11)

and

[picture] | (12)

Subtracting (12) from (11), we conclude the desired assertion.

Theorem 3.6 Consider that achieves (6) with Let be increasing such that and

[picture] | (13)

Then

Proof. Clearly that (13) is equivalent to being increasing. Define the formula

Since therefore, in view of Theorem 3.5, we obtain

[picture] | (14)

By Proposition 3.1, we have the following expression :

[picture] | (15)

Multiplying Eq.(15) by and replacing by , we arrive on

Consequently, we conclude that

By iterating the quantities and we attain in

But by utilizing Eq.(15) and Eq.(16), we find

Therefore, (14) becomes

[picture] | (16)

Hence the proof.

Theorem 3.7 Consider that achieves (6) with Let be decreasing such that and

[picture] | (17)

Then

Proof. By letting with the following properties:

•

•

•

The last property is valid by the monotonicity of in (17). Define a polynomial by utilizing as follows: for

where

satisfying

[picture] | (18)

Obviously, satisfies

This implies that is uniformly bounded on a compact set for By the definition of the Turan determinant, we obtain

where

such that

We conclude that there exists a constant such that

Remark 3.1 If in Theorems 3.6 and 3.7, we obtain that the coefficient For example, if (starlike class), then Thus, Theorem 3.7 implies that as Moreover, the above results can be considered for a sequence of polynomials

4. Applications

In this section, we utilize the Turan determinant to fined the coefficients bound of the classes and . We have the following propositions.

Proposition 4.1 Consider the classes and Then and

Proof. By utilizing and respectively for finding the upper bound of and A computation implies that

and

In view of Remark 3.1, we conclude that

Similarly for

In the similar manner of Proposition 4.1, we have the following result:

Proposition 4.2 Consider the classes Then

Proof. By utilizing we obtain

which implies that when

5. Conclusion

We imposed a new technique for finding the coefficients bound. This method based on several types of matrices. The major type was the Tura in the open unit disk. We proved the boundedness of this matrix from below as well as from above. We defined classes of analytic functions, depending on one coefficients, calculating by some special type of partial sums. The stability of these classes is considered by utilizing the Hurwitz matrix. We illustrated some applications of this method for two well defined classes (starlike and convex). The above method can be employed on other classes such as uniform, concave etc.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this article.

References

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[11]   | M. Darus, R. W. Ibrahim, Partial sums of analytic functions of bounded turning with applications. Comput. Appl. Math. 29(1), 81-88 (2010).
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