scholarly journals The Third-Order Hermitian Toeplitz Determinant for Alpha-Convex Functions

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1274
Author(s):  
Anna Dobosz
Keyword(s):  
Convex Functions ◽  
Upper Bounds ◽  
Arithmetic Mean ◽  
Lower And Upper Bounds ◽  
Third Order ◽  
Toeplitz Determinants ◽  
The Third ◽  
Toeplitz Determinant ◽  
Symmetry Properties ◽  
Definition Of

Sharp lower and upper bounds of the second- and third-order Hermitian Toeplitz determinants for the class of α-convex functions were found. The symmetry properties of the arithmetic mean underlying the definition of α-convexity and the symmetry properties of Hermitian matrices were used.

scholarly journals Third-Order Hankel and Toeplitz Determinants for Starlike Functions Connected with the Sine Function

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 404 ◽  
Author(s):  
Hai-Yan Zhang ◽  
Rekha Srivastava ◽  
Huo Tang
Keyword(s):  
Starlike Functions ◽  
Function Class ◽  
Open Unit ◽  
Sine Function ◽  
Hankel Determinant ◽  
Third Order ◽  
Toeplitz Determinants ◽  
The Third ◽  
Toeplitz Determinant ◽  
The Right

Let S s * be the class of normalized functions f defined in the open unit disk D = { z : | z | < 1 } such that the quantity z f ′ ( z ) f ( z ) lies in an eight-shaped region in the right-half plane and satisfying the condition z f ′ ( z ) f ( z ) ≺ 1 + sin z ( z ∈ D ) . In this paper, we aim to investigate the third-order Hankel determinant H 3 ( 1 ) and Toeplitz determinant T 3 ( 2 ) for this function class S s * associated with sine function and obtain the upper bounds of the determinants H 3 ( 1 ) and T 3 ( 2 ) .

Third Hankel determinants for two classes of analytic functions with real coefficients

2021 ◽  
Vol 33 (4) ◽  
pp. 973-986
Author(s):  
Young Jae Sim ◽  
Paweł Zaprawa
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Hankel Determinant ◽  
Lower And Upper Bounds ◽  
Hankel Determinants ◽  
The Third ◽  
Typically Real ◽  
Class Of Functions ◽  
Typically Real Functions

Abstract In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

scholarly journals Bounds on the Third Order Hankel Determinant for Certain Subclasses of Analytic Functions

2017 ◽  
Vol 25 (3) ◽  
pp. 199-214
Author(s):  
S.P. Vijayalakshmi ◽  
T.V. Sudharsan ◽  
Daniel Breaz ◽  
K.G. Subramanian
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Analytic Functions ◽  
Unit Disc ◽  
Taylor Series Expansion ◽  
Upper Bounds ◽  
Hankel Determinant ◽  
Third Order ◽  
The Third

Abstract Let A be the class of analytic functions f(z) in the unit disc ∆ = {z ∈ C : |z| < 1g with the Taylor series expansion about the origin given by f(z) = z+ ∑n=2∞ anzn, z ∈∆ : The focus of this paper is on deriving upper bounds for the third order Hankel determinant H3(1) for two new subclasses of A.

Generalizing the McClelland Bounds for Total π-Electron Energy

2008 ◽  
Vol 63 (5-6) ◽  
pp. 280-282 ◽  
Author(s):  
Ivan Gutman ◽  
Gopalapillai Indulal ◽  
Roberto Todeschinic
Keyword(s):  
Electron Energy ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Arithmetic Mean ◽  
Lower And Upper Bounds ◽  
Real Numbers ◽  
Laplacian Energy ◽  
Graph Energy

In 1971 McClelland obtained lower and upper bounds for the total π-electron energy. We now formulate the generalized version of these bounds, applicable to the energy-like expression EX = Σni =1 |xi − x̅|, where x1,x2, . . . ,xn are any real numbers, and x̅ is their arithmetic mean. In particular, if x1,x2, . . . ,xn are the eigenvalues of the adjacency, Laplacian, or distance matrix of some graph G, then EX is the graph energy, Laplacian energy, or distance energy, respectively, of G.

A third order canonical approximation of the general solution of the restricted problem of three bodies in the neighbourhood of L4

1966 ◽  
Vol 25 ◽  
pp. 170-175
Author(s):  
A. Deprit
Keyword(s):  
General Solution ◽  
Restricted Problem ◽  
Body Problem ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Third Order ◽  
The Third ◽  
Transformation Of Variables ◽  
Definition Of ◽  
Three Body

A canonical transformation of variables is introduced in the plane restricted three-body problem which gives the Hamiltonian in the form of a power series with normalized second order terms. Then a generating function is constructed, step by step, that permits the definition of new action and angle variables, such that the Hamiltonian is independent of the angle variables. This procedure has been done explicitly up to the third order terms.

Polynomial decay of correlations for almost Anosov diffeomorphisms

2017 ◽  
Vol 39 (3) ◽  
pp. 832-864
Author(s):  
XU ZHANG ◽  
HUYI HU
Keyword(s):  
Fixed Point ◽  
Decay Of Correlations ◽  
Polynomial Decay ◽  
Lower And Upper Bounds ◽  
Third Order ◽  
Anosov Diffeomorphisms ◽  
The Third ◽  
Srb Measures ◽  
Taylor Expansions ◽  
Almost Anosov

We investigate the polynomial lower and upper bounds for decay of correlations of a class of two-dimensional almost Anosov diffeomorphisms with respect to their Sinai–Ruelle–Bowen (SRB) measures. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the indifferent fixed point, and are expressed by using coefficients of the third-order terms in the Taylor expansions of the diffeomorphisms at the indifferent fixed point.

scholarly journals XX. A memoir on curves of the third order

1857 ◽  
Vol 147 ◽  
pp. 415-446 ◽  
Keyword(s):  
Class I ◽  
Geometrical Theory ◽  
Third Order ◽  
The Third ◽  
Cubic Curve ◽  
The Subject ◽  
Definition Of

A curve of the third order, or cubic curve, is the locus represented by an equation such as U=(*)( x , y , z ) 3 =0; and it appears by my “Third Memoir on Quantics,” that it is proper to consider, in connexion with the curve of the third order U = 0, and its Hessian HU=0 (which is also a curve of the third order), two curves of the third class, viz. the curves represented by the equations PU=0 and QU=0. These equations, I say, represent curves of the third class; in fact, PU and QU are contravariants of U, and therefore, when the variables x , y , z of U are considered as point coordinates, the variables ξ, η, ζ of PU and QU must be considered as line coordinates, and the curves will be curves of the third class. I propose (in analogy with the form of the word Hessian) to call the two curves in question the Pippian and Quippian respectively. A geometrical definition of the Pippian was readily found; the curve is in fact Steiner’s curve R 0 mentioned in the memoir “Allgemeine Eigenschaften der algebraischen Curven,” Crelle , t. xlvii. pp. 1-6, in the particular case of a basis-curve of the third order; and I also found that the Pippian might be considered as occurring implicitly in my “Mémoire sur les Courbes du Troisiéme Ordre,” Liouville , t. ix. p. 285, and “Nouvelles Remarques sur les Courbes du Troisiéme Ordre,” Liouville , t. x. p. 102. As regards the Quippian, I have not succeeded in obtaining a satisfactory geometrical definition; but the search after it led to a variety of theorems, relating chiefly to the first-mentioned curve, and the results of the investigation are contained in the present memoir. Some of these results are due to Mr. Salmon, with whom I was in correspondence on the subject. The character of the results makes it diflicult to develope them in a systematic order; hut the results are given in such connexion one with another as I have been able to present them in. Considering the object of the memoir to be the establishment of a distinct geometrical theory of the Pippian, the leading results will be found summed up in the nine different definitions or modes of generation of the Pippian, given in the concluding number. In the course of the memoir I give some further developments relating to the theory in the memoirs in Liouville above referred to, showing its relation to the Pippian, and the analogy with theorems of Hesse in relation to the Hessian. Article No. 1.— Definitions , &c . 1. It may be convenient to premise as follows:—Considering, in connexion with a curve of the third order or cubic, a point , we have— ( a ) The first or conic polar of the point. ( b ) The second or line polar of the point. The meaning of these terms is well known, and they require no explanation.

The Finite Screw System Associated With the Displacement of a Line

2003 ◽  
Vol 125 (1) ◽  
pp. 105-109 ◽  
Author(s):  
Chintien Huang ◽  
Jin-Cheng Wang
Keyword(s):  
Exceptional Case ◽  
The Other ◽  
Third Order ◽  
Analytical Geometry ◽  
The Third ◽  
Linearly Independent ◽  
Finite Displacement ◽  
Definition Of

In determining the screw systems associated with incompletely specified displacements, the displacement of a line was known to be an exceptional case. Recent research has concluded that all possible screws for the finite displacement of a line do not form a screw system. This paper utilizes Dimentberg’s definition of pitch to demonstrate that all possible screws for displacing a line from one position to another can indeed form a screw system of the third order. Two different approaches are taken: one uses the concept of a screw triangle, and the other is based on analytical geometry. A set of three linearly independent screws of the screw system is shown to be perpendicularly intersecting the external bisector of the initial and final positions of the line.

scholarly journals Inequalities for Jensen–Sharma–Mittal and Jeffreys–Sharma–Mittal Type f–Divergences

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1688
Author(s):  
Paweł A. Kluza
Keyword(s):  
Convex Functions ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
New Inequalities

In this paper, we introduce new divergences called Jensen–Sharma–Mittal and Jeffreys–Sharma–Mittal in relation to convex functions. Some theorems, which give the lower and upper bounds for two new introduced divergences, are provided. The obtained results imply some new inequalities corresponding to known divergences. Some examples, which show that these are the generalizations of Rényi, Tsallis, and Kullback–Leibler types of divergences, are provided in order to show a few applications of new divergences.

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