Abstract
This article constructs trigonometric polynomials of the sine and cosine whose sums are nonnegative. As an application, those nonnegative trigonometric sums are used to study the geometric properties of complex polynomials in the unit disk. The Strum sequences are used to prove the main outcome.
1. Introduction
The trigonometric polynomials with nonnegative sums and its enormous application in various branches of mathematics and science are well known. There are several theories in the literature to find necessary (or/and sufficient) conditions by whichare nonnegative or positive. This is a century old problem and many mathematicians [1–10] have contributed numerous results. The geometric function theory (univalent functions) has a very close association with the positive trigonometric sums. Both areas have taken and given to each other as evident in the work [11–22]. Here, we intend to present few more results of this interplay.
An analytic function is univalent in the unit disk if for implies . Let be the class of all univalent functions with normalization and have the series form
Denote by , the class of all analytic functions of form (2). A domain is said to be starlike with respect to a point if the line segment joining any arbitrary chosen points in with lies completely in . The domain will be convex if it is star-like with respect to all of its points. Analytically, these classes are characterized aswhere and , respectively, denote the class of star-like and convex functions.
The aim of this article is to construct the star-like polynomials. It is known that partial sum of a univalent function in need not be univalent. For example, the Koebe function is an extremal function of many classes in the univalent functions theory defined on . However, it can be observed that , the partial sum of the Taylor series for is univalent only on the disk . Thus, the geometric properties of an analytic function do not inherit to its partial sum. These motivate us to construct the polynomial which are star-like on .
The following result related to the starlikeness of is required in sequel.
Lemma 1. (see [23]). If is typically real in , that is, for and satisfies , , then is star-like in .
It is clear from Lemma 1 that the positivity of cosine and sine sums has a significant role to study the star-like polynomials. During this investigation, we observed that, along with other conditions, the monotonicity of is pivotal for the positivity or nonnegativity for the sine and cosine sums. This makes us think about what will be the case if we chose randomly (especially nonmonotone)? How we can prove the positivity of those trigonometric sums with arbitrary coefficients? Can those positive sums have an application in the study of geometric function theory?
Here, we will provide some affirmative answers for the above questions. The work is motivated from the recent work by Kwong [24], where several trigonometric polynomials with positive sums are given.
2. Statements of Main Results and Their Consequences
Now, we will state our main results. The first result will give the positivity of cosine and sine sum with nonmonotonic coefficients.
Theorem 1. For , the cosine sumand the sine sumare positive.
The basic identity implies that for all .
In the sense of complex function theory, the above two positive trigonometric sums can be described as follows. On the boundary of the unit circle , that is, for , ; while when . Now, by reflection principal, it follows that when . Here,
Denote . Then, clearly and , . Since , the minimum principal for harmonic functions yield ; similarly, when .
The above fact together with Lemma 1 gives the following result.
Theorem 2. The polynomialis star-like with respect to the origin in .
Theorem 1 can also be validated from Figures 1(a) and 1(b), while starlikeness of can be seen in Figure 2.
Now, by the same argument as in Theorem 1, several other examples can be constructed by suitable positive cosine and sine sums. Now, we state our second theorem.
(a)
(b)
Theorem 3. The polynomialmaps the unit disk to a star-like domain.
Our third result can be stated as follows.
Theorem 4. The polynomialmaps the unit disk to a star-like domain for
Before we give analytical Proof of Theorem 4 in Section 3, the starlikeness of and can be seen in Figures 3 and 4, respectively.
3. Proof of the Main Results
Proof of Theorem 1. To prove identity (1), we need to recall the classical identity:This impliesNow, we can rewrite (1) ashence proved.
Next, we will prove the positivity of the sine sum in (5). From the well-known trigonometric identities, one can obtain the following identities:It is known that is positive for , and hence ifDenote . Then, clearly and (14) is equivalent toSince , inequality (15) holds if have no zero in . We will prove these facts by constructing Strum sequences to determine the number of zeroes of a polynomial in an open interval.
The Strum sequence for the polynomial is given as follows:Here, by , we mean the remainder of the long division of by the polynomial . Now, by Strum theorem the number of roots of in is equal to , where is the number of change of sign by the Strum sequence at the point . From Table 1, it follows that . Thus, does not have any zero on .
Proof of Theorem 2. Clearly,Now, for , it follows thatand for ,The result follows from Lemma 1.
Proof of Theorem. 3. For , it follows from (3) thatfor .
Clearly, for , identity (20) givesSimilarly, for , we haveIf , then (22) implies that , while if , then . Thus, we can conclude that . Finally, the conclusion follows from Lemma 1.
Next, we will consider the case for . In this case,Similarly,It is well known that is positive for . Thus, it is enough to show thatwhich is equivalent to proveSince , we need to prove that does not have any zero in . We apply the Strum theorem for this purpose. A differentiation of (26) givesNext to form the Strum sequence, we need to find the remainder of and asand continuing, similar to the Strum theorem, we haveThe change of sign of the Strum sequence , respectively, at and can be seen in Table 2. Since , the polynomial does not have any zero on .
Finally, we have for all , and by Lemma 1, it can be concluded that is star-like with respect to the origin.
4. Concluding Remark
From the above three examples, it can be concluded that the trigonometric sine and cosine sum can also be positive with some nonmonotone coefficients.
In Theorem 4, though we prove that is star-like for , but the graphical experiment Figure 5 shows that is star-like for all . However, is negative for some range of when is larger (greater than 28, see Figure 6). Thus, our method is not applicable on for larger .
The results in this article indicate that several polynomials can be constructed which maps unit disk to a star-like domain by using positivity of cosine and sine sum.
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
T. Abdeljawad would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group no. RG-DES-2017-01-17.