scholarly journals Generalized typically real functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1697-1710 ◽  
Author(s):  
Stanisława Kanas ◽  
Anna Tatarczak
Keyword(s):  
Unit Disk ◽  
Real Function ◽  
Chebyshev Polynomials ◽  
Regular Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Functions ◽  
Typically Real ◽  
Necessary And Sufficient ◽  
Typically Real Functions

Let f(z)=z+a2z2+... be regular in the unit disk and real valued if and only if z is real and |z| < 1. Then f(z) is said to be typically real function. Rogosinski found the necessary and sufficient condition for a regular function to be typically-real. The main purpose of the paper is a consideration of the generalized typically-real functions defined via the generating function of the generalized Chebyshev polynomials of the second kind ?p,q(ei?;z)=1 /(1-pzei?)(1-qze-i?) = ??,n=0 Un(p,q; ei?)zn, where -1 ? p,q ? 1; ?? ?0,2??i, |z|<1.

Coefficient inequalities related with typically real functions

2020 ◽  
Vol 70 (4) ◽  
pp. 829-838
Author(s):  
Saqib Hussain ◽  
Shahid Khan ◽  
Khalida Inayat Noor ◽  
Mohsan Raza
Keyword(s):  
Unit Disk ◽  
Real Functions ◽  
Coefficient Inequalities ◽  
Typically Real ◽  
Class Of Functions ◽  
Typically Real Functions

AbstractIn this paper, we are mainly interested to study the generalization of typically real functions in the unit disk. We study some coefficient inequalities concerning this class of functions. In particular, we find the Zalcman conjecture for generalized typically real functions.

scholarly journals On the asymptotic boundary behavior of functions analytic in the unit disk

1977 ◽  
Vol 67 ◽  
pp. 1-13
Author(s):  
James R. Choike
Keyword(s):  
Riemann Surface ◽  
Analytic Function ◽  
Unit Disk ◽  
Boundary Behavior ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Asymptotic Boundary ◽  
Necessary And Sufficient ◽  
Class Of Functions

In [8] a necessary and sufficient condition was given for determining the equivalence of two asymptotic boundary paths for an analytic function w = f(p) on a Riemann surface F. In this paper we give a necessary and sufficient condition for determining the nonequivalence of two asymptotic boundary paths for f(z) analytic in |z| < R, 0 < R ≤ + ∞. We shall, also, illustrate some applications of the main result and examine a class of functions introduced by Valiron.

scholarly journals The complex geometry of Blaschke products of degree 3 and associated ellipses

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 61-68
Author(s):  
Masayo Fujimuraa
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Geometry ◽  
Sufficient Condition ◽  
Blaschke Products ◽  
Necessary And Sufficient Condition ◽  
Geometrical Properties ◽  
Natural Extensions ◽  
Necessary And Sufficient

A bicentric polygon is a polygon which has both an inscribed circle and a circumscribed one. For given two circles, the necessary and sufficient condition for existence of bicentric triangle for these two circles is known as Chapple?s formula or Euler?s theorem. As one of natural extensions of this formula, we characterize the inscribed ellipses of a triangle which is inscribed in the unit circle. We also discuss the condition for the ?circumscribed? ellipse of a triangle which is circumscribed about the unit circle. For the proof of these results, we use some geometrical properties of Blaschke products on the unit disk.

scholarly journals Composition operators fromℬαtoQKtype spaces

2008 ◽  
Vol 6 (1) ◽  
pp. 88-104 ◽  
Author(s):  
Jizhen Zhou
Keyword(s):  
Unit Disk ◽  
Composition Operator ◽  
Analytic Functions ◽  
Composition Operators ◽  
Nondecreasing Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Of Analytic Functions ◽  
Type Spaces ◽  
Necessary And Sufficient

Suppose thatϕis an analytic self-map of the unit diskΔ. Necessary and sufficient condition are given for the composition operatorCϕf=fοϕto be bounded and compact fromα-Bloch spaces toQKtype spaces which are defined by a nonnegative, nondecreasing functionk(r)for0≤r<∞. Moreover, the compactness of composition operatorCϕfromℬ0toQKtype spaces are studied, whereℬ0is the space of analytic functions offwithf′∈H∞and‖f‖ℬ0=|f(0)|+‖f′‖∞.

scholarly journals On typically real functions which are generated by a fixed typically real function

2011 ◽  
Vol 61 (3) ◽  
pp. 733-742
Author(s):  
Magdalena Sobczak-Kneć ◽  
Katarzyna Trąbka-Więcław
Keyword(s):  
Real Function ◽  
Real Functions ◽  
Typically Real ◽  
Typically Real Functions

On Some Classes of Univalent Polynomials

1978 ◽  
Vol 30 (02) ◽  
pp. 332-349 ◽  
Author(s):  
Q. I. Rahman ◽  
J. Szynal
Keyword(s):  
Unit Disk ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Image Position ◽  
Necessary And Sufficient

It was in the year 1931 that Dieudonné [4] proved the following necessary and sufficient condition for a polynomial to be univalent in the unit disk. THEOREM A (Dieudonné criterion). The polynomial is univalent in |z| &lt; 1 if and only if for every θ in [0, π/2] the associated polynomial does not vanish in |z| &lt; 1. For θ = 0, ϕ(z, θ) is to be interpreted as Pn'(z).

scholarly journals A covering theorem for odd typically-real functions

1980 ◽  
Vol 3 (1) ◽  
pp. 189-192
Author(s):  
E. P. Merkes
Keyword(s):  
Analytic Function ◽  
Real Function ◽  
Covering Theorem ◽  
Real Functions ◽  
Typically Real ◽  
One To One ◽  
Typically Real Functions

An analytic functionf(z)=z+a2z2+…in|z|<1is typically-real ifImf(z)Imz≥0. The largest domainGin which each odd typically-real function is univalent (one-to-one) and the domain⋂f(G)for all odd typically real functionsfare obtained.

scholarly journals Second Hankel Determinants for the Class of Typically Real Functions

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Paweł Zaprawa
Keyword(s):  
Unit Disk ◽  
Lower Bounds ◽  
Analytic Functions ◽  
Upper And Lower Bounds ◽  
Hankel Determinants ◽  
Real Functions ◽  
Typically Real ◽  
Typically Real Functions

We discuss the Hankel determinantsH2(n)=anan+2-an+12for typically real functions, that is, analytic functions which satisfy the conditionIm ⁡z Im⁡f(z)≥0in the unit disk Δ. Main results are concerned withH2(2)andH2(3). The sharp upper and lower bounds are given. In general case, forn≥4, the results are not sharp. Moreover, we present some remarks connected with typically real odd functions.

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