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 A Subfamily of Univalent Functions Associated with q-Analogue of Noor Integral Operator

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Muhammad Arif ◽  
Miraj Ul Haq ◽  
Jin-Lin Liu
Keyword(s):  
Integral Operator ◽  
Integral Representation ◽  
Linear Combination ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Coefficient Estimates ◽  
Arithmetic Means ◽  
Radius Of Starlikeness

The main objective of the present paper is to define a new subfamily of analytic functions using subordinations along with the newly defined q-Noor integral operator. We investigate a number of useful properties such as coefficient estimates, integral representation, linear combination, weighted and arithmetic means, and radius of starlikeness for this class.

IV.—On Least Squares and Linear Combination of Observations

1936 ◽  
Vol 55 ◽  
pp. 42-48 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Least Squares ◽  
Linear Combination ◽  
Mean Square Error ◽  
Mean Square ◽  
Approximate Representation ◽  
Linear Combinations ◽  
The Mean ◽  
Image Position

In a series of papers W. F. Sheppard (1912, 1914) has considered the approximate representation of equidistant, equally weighted, and uncorrelated observations under the following assumptions:–(i) The data beingu1, u2, …, un, the representation is to be given by linear combinations(ii) The linear combinations are to be such as would reproduce any set of values that were already values of a polynomial of degree not higher than thekth.(iii) The sum of squared coefficientswhich measures the mean square error ofyi, is to be a minimum for each value ofi.

scholarly journals Certain classes of univalent functions with negative coefficients II

1976 ◽  
Vol 15 (3) ◽  
pp. 467-473 ◽  
Author(s):  
V.P. Gupta ◽  
P.K. Jain
Keyword(s):  
Univalent Functions ◽  
Arithmetic Mean ◽  
Distortion Theorem ◽  
Linear Combinations ◽  
Radius Of Convexity ◽  
Image Position ◽  
Class Of Functions

Let P*(α, β) denote the class of functionsanalytic and univalent in |z| < 1 for whichwhere α є [0, 1), β є (0, 1].Sharp results concerning coefficients, distortion theorem and radius of convexity for the class P*(α, β) are determined. A comparable theorem for the classes C*(α, β) and P*(α, β) is also obtained. Furthermore, it is shown that the class P*(α, ß) is closed under ‘arithmetic mean’ and ‘convex linear combinations’.

scholarly journals Radius of starlikeness for some classes containing non-univalent functions

2021 ◽  
pp. 2250009
Author(s):  
Shalu Yadav ◽  
Kanika Sharma ◽  
V. Ravichandran
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Univalent Function ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Radius Of Starlikeness ◽  
The Past ◽  
The Right ◽  
Starlike Univalent Function

A starlike univalent function [Formula: see text] is characterized by [Formula: see text]; several subclasses of starlike functions were studied in the past by restricting [Formula: see text] to take values in a region [Formula: see text] on the right-half plane, or, equivalently, by requiring [Formula: see text] to be subordinate to the corresponding mapping of the unit disk [Formula: see text] to the region [Formula: see text]. The mappings [Formula: see text], [Formula: see text], defined by [Formula: see text] and [Formula: see text] map the unit disk [Formula: see text] to certain nice regions in the right-half plane. For normalized analytic functions [Formula: see text] with [Formula: see text] and [Formula: see text] are subordinate to the function [Formula: see text] for some analytic functions [Formula: see text] and [Formula: see text], we determine the sharp radius for them to belong to various subclasses of starlike functions.

scholarly journals On analytic functions with reference to an integral operator

1983 ◽  
Vol 28 (2) ◽  
pp. 207-215 ◽  
Author(s):  
R. Parvatham ◽  
T.N. Shanmugam
Keyword(s):  
Integral Operator ◽  
Analytic Functions ◽  
Sharp Estimate ◽  
Radius Of Starlikeness ◽  
Image Position ◽  
Class Of Functions

Let E = {z: |z| < 1} and let H = {w : regular in E, w(0) = 0, |w(z)| < l, z ∈ E}.Let P(A, B) denote the class of functions in E which can be put in the form (1 + Aw(z))/(1 + Bw(z)), −1 ≤ A < B ≤ 1, w(z) ∈ H. Let S*(A, B) denote the class of functions f(z) of the form such that zf′(z)/f(z) ∈ P(A, B). If f(z) ∈ S*(A, B) and g(z) ∈ S*(C, D) then, in this paper the radius of starlikeness of order β (β ∈ [0, 1]) of the following integral operatoris determined. Conversely, a sharp estimate is obtained for the radius of starlikeness of the class of functionswhere g(z) and F(z) belong to the class S*(A, B).

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 ∊ ).

scholarly journals Some applications of generalized Ruscheweyh derivatives involving a general fractional derivative operator to a class of analytic functions with negative coefficients II

2020 ◽  
Vol 28 (1) ◽  
pp. 85-103
Author(s):  
Waggas Galib Atshan ◽  
S. R. Kulkarni
Keyword(s):  
Fractional Derivative ◽  
Linear Combination ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Integral Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Derivative Operator ◽  
Necessary And Sufficient

AbstractIn this paper, we study a class of univalent functions f as defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator, satisfying{\mathop{\rm Re}\nolimits} \left\{{{{z\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)'} \over {\left({1 - \gamma} \right){\bf{J}}_1^{\lambda,\mu}f\left(z \right) + \gamma {z^2}\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)''}}} \right\} > \beta.A necessary and sufficient condition for a function to be in the class A_\gamma ^{\lambda,\mu,\nu}\left({n,\beta} \right) is obtained. Also, our paper includes linear combination, integral operators and we introduce the subclass A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right) consisting of functions with negative and fixed finitely many coefficients. We study some interesting properties of A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right).

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.

scholarly journals On a functional equation for the exponential function of a complex variable

1971 ◽  
Vol 12 (1) ◽  
pp. 31-34 ◽  
Author(s):  
Hiroshi Haruki
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Exponential Function ◽  
Complex Variable ◽  
Analytic Functions ◽  
Real Constant ◽  
Complex Constant ◽  
Arbitrary Real Constant ◽  
Arbitrary Complex ◽  
Image Position

The following result is well known in the theory of analytic functions; see [1].Theorem A. Suppose that f(z) is an entire function of a complex variable z. Then f(z) satisfies the functional equationwhere z = x + iy (x, y real), if and only if f(z) = aexp(sz), where a is an arbitrary complex constant and s is an arbitrary real constant.

Starlike Univalent Functions Bounded on the Real Axis

1989 ◽  
Vol 41 (4) ◽  
pp. 642-658
Author(s):  
Richard Fournier
Keyword(s):  
Real Axis ◽  
Analytic Functions ◽  
Unit Disc ◽  
Univalent Functions ◽  
Uniform Norm ◽  
Open Unit ◽  
Open Unit Disc ◽  
The Real ◽  
Typically Real ◽  
Image Position

We denote by E the open unit disc in C and by H(E) the class of all analytic functions f on E with f(0) = 0. Let (see [3] for more complete definitions)S = {ƒ ∈ H(E)|ƒ is univalent on E}S0 = {ƒ ∈ H(E)|ƒ is starlike univalent on E}TR = {ƒ ∈ H(E)|ƒ is typically real on E}.The uniform norm on (— 1, 1) of a function ƒ ∈ H(E) is defined by

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