scholarly journals Two Points Taylor’s Type Representations for Analytic Complex Functions with Integral Remainders

2021 ◽  
Vol 29 (2) ◽  
pp. 131-154
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Convex Domain ◽  
Error Bounds ◽  
Analytic Functions ◽  
Complex Functions ◽  
Complex Exponential ◽  
Provided Examples

Abstract In this paper we establish some two point weighted Taylor’s expansions for analytic functions f : D ⊆ ℂ→ ℂ defined on a convex domain D. Some error bounds for these expansions are also provided. Examples for the complex logarithm and the complex exponential are also given.

Linear Combinations of Univalent Functions with Complex Coefficients

1971 ◽  
Vol 23 (4) ◽  
pp. 712-717 ◽  
Author(s):  
Robert K. Stump
Keyword(s):  
Linear Combination ◽  
Convex Domain ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Linear Combinations ◽  
Radius Of Starlikeness ◽  
Complex Constant ◽  
Complex Functions ◽  
Image Position ◽  
Complex Coefficients

Let U be the class of all normalized analytic functionswhere z ∈ E = {z : |z| < 1} and ƒ is univalent in E. Let K denote the sub-class of U consisting of those members that map E onto a convex domain. MacGregor [2] showed that if ƒ1 ∈ K and ƒ2 ∈ K and if1then F ∉ K when λ is real and 0 < λ < 1, and the radius of univalency and starlikeness for F is .In this paper, we examine the expression (1) when ƒ1 ∈ K, ƒ2 ∈ K and λ is a complex constant and find the radius of starlikeness for such a linear combination of complex functions with complex coefficients.

scholarly journals Errors of Gauss-Radau and Gauss-Lobatto quadratures with double end point

2019 ◽  
Vol 13 (2) ◽  
pp. 463-477
Author(s):  
Aleksandar Pejcev ◽  
Ljubica Mihic
Keyword(s):  
Explicit Expression ◽  
Error Bounds ◽  
Analytic Functions ◽  
Remainder Term ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
End Points ◽  
End Point ◽  
Appl Math

Starting from the explicit expression of the corresponding kernels, derived by Gautschi and Li (W. Gautschi, S. Li: The remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadrature rules with multiple end points, J. Comput. Appl. Math. 33 (1990) 315{329), we determine the exact dimensions of the minimal ellipses on which the modulus of the kernel starts to behave in the described way. The effective error bounds for Gauss- Radau and Gauss-Lobatto quadrature formulas with double end point(s) are derived. The comparisons are made with the actual errors.

scholarly journals Harmonic close-to-convex mappings

2002 ◽  
Vol 15 (1) ◽  
pp. 23-28 ◽  
Author(s):  
Jay M. Jahangiri ◽  
Herb Silverman
Keyword(s):  
Analytic Functions ◽  
Harmonic Functions ◽  
Convex Mappings ◽  
Complex Functions

Sufficient coefficient conditions for complex functions to be close-to-convex harmonic or convex harmonic are given. Construction of close-to-convex harmonic functions is also studied by looking at transforms of convex analytic functions. Finally, a convolution property for harmonic functions is discussed.

scholarly journals Differential Subordinations for Nonanalytic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Georgia Irina Oros ◽  
Gheorghe Oros
Keyword(s):  
Complex Plane ◽  
Classical Theory ◽  
Analytic Functions ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Differential Subordination ◽  
Analytic Case ◽  
Complex Functions ◽  
Differential Subordinations

In the paper by Mocanu (1980), Mocanu has obtained sufficient conditions for a function in the classesC1(U), respectively, andC2(U)to be univalent and to mapUonto a domain which is starlike (with respect to origin), respectively, and convex. Those conditions are similar to those in the analytic case. In the paper by Mocanu (1981), Mocanu has obtained sufficient conditions of univalency for complex functions in the classC1which are also similar to those in the analytic case. Having those papers as inspiration, we try to introduce the notion of subordination for nonanalytic functions of classesC1andC2following the classical theory of differential subordination for analytic functions introduced by Miller and Mocanu in their papers (1978 and 1981) and developed in their book (2000). LetΩbe any set in the complex planeC, letpbe a nonanalytic function in the unit discU,p∈C2(U),and letψ(r,s,t;z):C3×U→C. In this paper, we consider the problem of determining properties of the functionp, nonanalytic in the unit discU, such thatpsatisfies the differential subordinationψ(p(z),Dp(z),D2p(z)-Dp(z);z)⊂Ω⇒p(U)⊂Δ.

scholarly journals Function theory for a beltrami algebra

1985 ◽  
Vol 8 (2) ◽  
pp. 247-256
Author(s):  
B. A. Case
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic System ◽  
Analytic Functions ◽  
Function Theory ◽  
Real Parameter ◽  
Partial Differential ◽  
Complex Functions ◽  
Parameter Function

Complex functions are investigated which are solutions of an elliptic system of partial differential equations associated with a real parameter function. The functionsfassociated with a particualr parameter functiongon a domainDform a Beltrami algebra denoted by the pair(D,g)and a function theory is developed in this algebra. A strong conformality property holds for all functions in a(D,g)algebra. Forg≡|z|=rthe algebra(D,r)is that of the analytic functions.

scholarly journals Error bounds for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 231-239 ◽  
Author(s):  
Ljubica Mihic ◽  
Aleksandar Pejcev ◽  
Miodrag Spalevic
Keyword(s):  
Weight Function ◽  
Quadrature Formula ◽  
Error Bounds ◽  
Analytic Functions ◽  
Remainder Term ◽  
Applied Mathematics ◽  
Maximum Modulus ◽  
End Points ◽  
The Third ◽  
Fourth Kind

For analytic functions the remainder terms of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points -+1, for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi and Li, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi and Li in paper [The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points, Journal of Computational and Applied Mathematics 33 (1990) 315-329.]

scholarly journals The Closed Ideals in an Algebra of Analytic Functions

1957 ◽  
Vol 9 ◽  
pp. 426-434 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Banach Algebra ◽  
Complex Plane ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Complex Functions ◽  
Continuous Complex ◽  
Image Position

Let K and C be the closure and boundary, respectively, of the open unit disc U in the complex plane. Let be the Banach algebra whose elements are those continuous complex functions on K which are analytic in U, with norm (f ∊ ).

Sign in / Sign up

Export Citation Format

Share Document