On the Eneström-Kakeya Theorem

In $[4,7]$, Enestrom Kakeya theorem has stated as the following. If $f(z)=\sum_{j=0}^n k_j z^j$ is the $n^{t h}$ degree polynomial with real coefficients such that $0<k_0 \leq k_1 \leq \ldots \leq k_{n-2} \leq k_{n-1} \leq k_n$ then all zeros of $\mathrm{f}(\mathrm{z})$ lies in $|z| \leq 1$. In [1], Aziz and Mahammad, showed that zeros of $f(z)$ satisfies $|z| \geq \frac{n}{n+1}$ are simple, under the same conditions. In this paper, we extend the above result to the polar derivative by relaxing the hypothesis in different ways.

We prove some extensions of the classical results concerning the Enestrom-Kakeya theorem and related analytic functions. Besides several consequences, our results considerably improve the bounds by relaxing and weakening the hypothesis in some cases.

Chapter 1. Boundary Behavior of Univalent Harmonic Mappings: A Survey of Recent Boundary Behavior Results of Univalent Harmonic Mappings.- Chapter 2. Harmonic Univalent Mappings and Minimal Graphs.- Chapter 3. Minimal Surfaces over the Slanted Half Planes, Vertical Strips and Single Slit.- Chapter 4. A Survey on Special Classes of Bazilevic Functions and Related Function Classes.- Chapter 5. Uniqueness of Entire Functions Sharing Certain Values with Derivatives.- Chapter 6. Differential Superordinations and Sandwich-Type Results.- Chapter 7. Starlikeness and Convexity of Certain Integral Transforms by using Duality Technique.- Chapter 8. Enestrom-Kakeya Theorem and Some of Its Generalizations.- Chapter 9. Starlikeness, Convexity and Close-to-Convexity of Harmonic Mappings.- Chapter 10. On Generalized p-Valent Non-Bazilevic Type Functions.- Chapter 11. Integral Mean Estimates for a Polynomial with Restricted Zeros.- Chapter 12. Uniqueness Results for Meromorphic Functions Concerning Small Functions.- Chapter 13. Maxima Polynomial Ranges for a Domain of Intersection of Two Circular Disks.

The most basic and important theorem concerning polynomials is the Fundamental Theorem of Algebra. This theorem, which tells us that every polynomial factors completely over the complex numbers, is the starting point for this book. Some of the intricate relationships between the location of the zeros of a polynomial and its coefficients are explored in Section 2. The equally intricate relationships between the zeros of a polynomial and the zeros of its derivative or integral are the subject of Section 1.3. This chapter serves as a general introduction to the body of theory known as the geometry of polynomials. Highlights of this chapter include the Fundamental Theorem of Algebra, the Enestrom-Kakeya theorem, Lucas’ theorem, and Walsh’s two-circle theorem.

In this paper we present certain interesting refinements of a well-known Enestrom-Kakeya theorem in the theory of distribution of zeros of polynomials which among other things also improve upon some results of Aziz and Mohammad, Govil and Rehman and others.

In this paper, we prove some generalizations of results concerning the Enestrm-Kakeya theorem. The results obtained considerably improve the bounds by relaxing the hypothesis in some cases.

By considering a multiply connected compact set \(G \subset \mathbb{C}\) and an analytic function on G, we attach the q-Bernstein–Faber–Walsh polynomials with q ≥ 1, for which Voronovskaja-type results with quantitative upper estimates are given and the exact orders of approximation in G for these polynomials, namely \(\frac{1} {n}\) if q = 1 and \(\frac{1} {q^{n}}\) if q > 1, are obtained. Also, given a sequence with the property λ n ↘ 0 as fast as we want, a type of Szasz–Mirakjan–Faber–Walsh operator is attached to G, for which the approximation order O(λ n ) is proved. The results are generalizations of those previously obtained by the author for the q-Bernstein–Faber polynomials and Szasz–Faber type operators attached to simply connected compact sets of the complex plane. The proof of existence for the Faber–Walsh polynomials used in our constructions is strongly based on some results on the location of critical points obtained in the book of Walsh (The location of critical points of analytic and harmonic functions, vol 34. American Mathematical Society, New York, 1950), which is also used in the major book of Rahman–Schmeisser (Analytic theory of polynomials, vol 26. Oxford University Press Inc, New York, 2002). At the end of the chapter, we present and motivate a conjecture and an open question concerning the use of truncated classical Szasz–Mirakjan operators in weighted approximation and in solving a generalization of the Szego’s problem concerning the zeroes distribution for the partial sums of the exponential function, respectively. Concerning the open question, the extensions of Enestrom–Kakeya Theorem in Govil–Rahman (Tohoku Math J 20(2):126–136, 1968) and other results on the location of the zeroes of polynomials in the Rahman–Schmeisser’s book (Analytic theory of polynomials, vol 26. Oxford University Press Inc, New York, 2002) are of interest.

In this paper, we prove a more general result concerning the location of the zeros of a polynomial in a ring shaped region from which we deduce an interesting and significant refinement of a classical result of Cauchy.A variety of other results, which in particular include several known extensions and generalizations of Enestrom -Kakeya Theorem, can be established from this result by a fairly uniform procedure.

According to the Enestrom-Kakeya theorem “zeros of the polynomial whose coefficients are positive, real and increasing along with the powers of the variable are lie in the unit circle” see [6, 10]. In [1], Aziz and Mahammad, showed that zeros of f(z) satisfies |z| ≥ n/n+1 are simple, under the same conditions. This article shows that the result of Gulzar, Zargar and Akthar in [8] is simplified in terms of real and imaginary parts of complex coefficients of the polynomial, also it extends some generalizations by imposing conditions on hypothesis in different ways.

In this paper we prove some results concerning the distribution of the zeros of a polynomial in the complex plane.Our results not only contain some known generalizations of Eneström-Kakeya theorem but also a variety of interesting results can be deduced from them by a fairly uniform procedure.

The classical Enestrom-Kakeya theorem establishes upper and lower bounds on the zeros of a polynomial with positive coefficients that are explicit functions of those coefficients. We establish a unifying framework that incorporates this theorem and several similar ones as special cases, while generating new theorems of a similar type. These establish zero inclusion and exclusion regions consisting of a single disk or the union of several disks in the complex plane. Our framework is built on two basic tools, namely a generalization of an observation by Cauchy, and a family of polynomial multipliers. Its approach is transparent and reduces algebraic manipulations to a minimum.

The problem of finding or characterizing the zeros of polynomial functions has a long history. The obtained results varied from theory, algorithms and numerical simulations. Historically, this study began with the fundamental theorem of algebra proved by Gauss. The most known results in this direction is the fact that a real polynomial of degree n has at most n real zeros. There is also the so called Descartes rule of signs concerning the number of positive zeros. Some generalizations of Descartes rule are know such as the BudanFourier theorem that gives an upper bound for the number of zeros of a polynomial. Also, the Sturm’s theorem that gives a method for determining the exact number of zeros in an interval [Hen, chapter 6] and [Hou, chapter 2]. Recent results uses the classical Enestrom-Kakeya theorem to restricts the location of the zeros based on a condition imposed on the coefficients of the polynomial under invistigation. See [Bre] and references therain. In this paper, we will use a dynamical system result concerning periodic points of a continuous function. The result is called Sharkovsky theorem [Sha1, Sha2, Sha3, Sha4, Sha5] that gives a complete description of possible sets of periods for continuous mappings defined on an interval. The interval need not be closed or bounded. The main idea used here is the notion of

In this paper, we prove some extensions and generalizations of the classical Enestrom-Kakeya theorem.

This paper surveys the zero distribution and inequalities for zeros of algebraic polynomials. Besides the basic facts on the zero distribution we consider the Grace’s theorem and many of its applications, the zero distribution for real polynomials, as well as the Enestrom-Kakeya theorem for a special class of polynomials. Also, we give some estimates for a number of zeros of a polynomial in a given domain in the complex plane.

For a polynomial of degree n, we have obtained an upper bound involving coefficients of the polynomial, for moduli of its p zeros of smallest moduli, and then a refinement of the well-known Enestrom-Kakeya theorem (under certain conditions).