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 A covering theorem for typically real functions

1969 ◽  
Vol 10 (2) ◽  
pp. 153-155
Author(s):  
D. A. Brannan ◽  
W. E. Kirwan
Keyword(s):  
Covering Theorem ◽  
Real Functions ◽  
Typically Real ◽  
Class Of Functions ◽  
Typically Real Functions

Let T denote the class of functionsf(z) = z+a2z2+…that are analytic in U = {|z| <1}, and satisfy the conditionImf(z). Imz≧ 0 (zεU).Thus T denotes the class of typically real functions introduced by W. Rogosinski [5].One of the most striking results in the theory of functionsg(z) = z + b2z2…that are analytic and univalent in U is the Koebe-Bieberbach covering theorem which states that {|w| <¼} ⊂ g(U). In this note we point out that the same result holds for functions in the class T, a fact which seems to have been overlooked previously. We also determine the largest subdomain of U in which every f(z) in T is univalent, extending previous results in [1] and [2].

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.

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.

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.

Coefficient Estimates of Some Classes of Analytic Functions

1983 ◽  
Vol 26 (2) ◽  
pp. 202-208
Author(s):  
Nicolas Samaris
Keyword(s):  
Analytic Functions ◽  
Positive Real Part ◽  
Coefficient Estimates ◽  
Positive Real ◽  
Real Functions ◽  
Typically Real ◽  
Typically Real Functions

AbstractWe are concerned with coefficient estimates, and other similar problems, of the typically real functions and of the functions with positive real part. Following the stream of ideas in [1], new results and generalizations of others given in [1], [2] and [3] are obtained.

On the Zalcman conjecture for starlike and typically real functions

1986 ◽  
Vol 191 (3) ◽  
pp. 467-474 ◽  
Author(s):  
Johnny E. Brown ◽  
Anna Tsao
Keyword(s):  
Real Functions ◽  
Typically Real ◽  
Typically Real Functions
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