scholarly journals Hankel Determinants for New Subclasses of Analytic Functions Related to a Shell Shaped Region

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1041 ◽  
Author(s):  
Gangadharan Murugusundaramoorthy ◽  
Teodor Bulboacă
Keyword(s):  
Power Series ◽  
Upper Bound ◽  
Analytic Functions ◽  
Series Expansions ◽  
Hankel Determinant ◽  
Hankel Determinants ◽  
Power Series Expansions

Using the operator L c a defined by Carlson and Shaffer, we defined a new subclass of analytic functions ML c a ( λ ; ψ ) defined by a subordination relation to the shell shaped function ψ ( z ) = z + 1 + z 2 . We determined estimate bounds of the four coefficients of the power series expansions, we gave upper bound for the Fekete–SzegőSzegő functional and for the Hankel determinant of order two for f ∈ ML c a ( λ ; ψ ) .

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 Parallel Software to Offset the Cost of Higher Precision

2021 ◽  
Vol 40 (2) ◽  
pp. 59-64
Author(s):  
Jan Verschelde
Keyword(s):  
Power Series ◽  
Parallel Algorithms ◽  
Series Expansions ◽  
Use Case ◽  
Double Precision ◽  
Algebraic Space ◽  
Space Curves ◽  
Computational Overhead ◽  
Power Series Expansions ◽  
The Cost

Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.

scholarly journals Coefficient inequalities for a class of analytic functions associated with the lemniscate of Bernoulli

2018 ◽  
Vol 37 (4) ◽  
pp. 83-95
Author(s):  
Trailokya Panigrahi ◽  
Janusz Sokól
Keyword(s):  
Upper Bound ◽  
Analytic Functions ◽  
Hankel Determinant ◽  
Lemniscate Of Bernoulli ◽  
Toeplitz Determinant ◽  
Second Hankel Determinant ◽  
Coefficient Inequalities ◽  
The Right

In this paper, a new subclass of analytic functions ML_{\lambda}^{*}  associated with the right half of the lemniscate of Bernoulli is introduced. The sharp upper bound for the Fekete-Szego functional |a_{3}-\mu a_{2}^{2}|  for both real and complex \mu are considered. Further, the sharp upper bound to the second Hankel determinant |H_{2}(1)| for the function f in the class ML_{\lambda}^{*} using Toeplitz determinant is studied. Relevances of the main results are also briefly indicated.

scholarly journals Free Function Theory Through Matrix Invariants

2017 ◽  
Vol 69 (02) ◽  
pp. 408-433 ◽  
Author(s):  
Igor Klep ◽  
Špela Špenko
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Function Theory ◽  
Rational Functions ◽  
Generalized Polynomials ◽  
Direct Sums ◽  
Real Analytic Functions ◽  
Power Series Expansions ◽  
Noncommutative Polynomials ◽  
Free Function

Abstract This paper concerns free function theory. Freemaps are free analogs of analytic functions in several complex variables and are defined in terms of freely noncommuting variables. A function of g noncommuting variables is a function on g-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions, and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps with involution, free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invarianttheoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involutionfree counterparts.

scholarly journals Coefficient estimates of certain subclasses of analytic functions associated with Hohlov operator

2019 ◽  
pp. 2150021
Author(s):  
P. Gochhayat ◽  
A. Prajapati ◽  
A. K. Sahoo
Keyword(s):  
Upper Bound ◽  
Analytic Functions ◽  
Gauss Hypergeometric Function ◽  
Open Unit ◽  
Sufficient Condition ◽  
Extremal Functions ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Coefficient Estimates ◽  
Extremal Value

A typical quandary in geometric functions theory is to study a functional composed of amalgamations of the coefficients of the pristine function. Conventionally, there is a parameter over which the extremal value of the functional is needed. The present paper deals with consequential functional of this type. By making use of Hohlov operator, a new subclass [Formula: see text] of analytic functions defined in the open unit disk is introduced. For both real and complex parameter, the sharp bounds for the Fekete–Szegö problems are found. An attempt has also been taken to found the sharp upper bound to the second and third Hankel determinant for functions belonging to this class. All the extremal functions are express in term of Gauss hypergeometric function and convolution. Finally, the sufficient condition for functions to be in [Formula: see text] is derived. Relevant connections of the new results with well-known ones are pointed out.

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