scholarly journals New distortion theorems for functions of positive real part and applications to the partial sums of univalent convex functions

1974 ◽  
Vol 45 (1) ◽  
pp. 113-113 ◽  
Author(s):  
S. D. Bernardi
Keyword(s):  
Convex Functions ◽  
Positive Real Part ◽  
Partial Sums ◽  
Positive Real ◽  
Distortion Theorems

scholarly journals On Mean Distortion for Analytic Functions with Positive Real Part in a Circle

1967 ◽  
Vol 29 ◽  
pp. 221-228
Author(s):  
Yûsaku Komatu
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Classical Theorem ◽  
Positive Real ◽  
Distortion Theorems

Let be the class of analytic functions Ф(z) which are regular and of positive real part in the unit circle | z | <1 and normalized by Ф(0) = 1. Several distortion theorems have been obtained on various functionals in this class. In a previous paper [4] we have dealt with mean distortion which generalizes a classical theorem of Rogosinski [6].

scholarly journals On Some Subclasses of Harmonic Functions Defined by Fractional Calculus

2008 ◽  
Vol 2008 ◽  
pp. 1-9
Author(s):  
R. A. Al-Khal ◽  
H. A. Al-Kharsani
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Harmonic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Sharp Estimates ◽  
Distortion Theorems

The purpose of this paper is to study subclasses of normalized harmonic functions with positive real part using fractional derivative. Sharp estimates for coefficients and distortion theorems are given.

scholarly journals On the Fekete–Szegö Type Functionals for Close-to-Convex Functions

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1497 ◽  
Author(s):  
Katarzyna Tra̧bka-Wiȩcław ◽  
Paweł Zaprawa ◽  
Magdalena Gregorczyk ◽  
Andrzej Rysak
Keyword(s):  
Analytic Function ◽  
Real Number ◽  
Convex Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Hankel Determinants ◽  
Coefficient Problems ◽  
Theoretical Results

In this paper, we consider two functionals of the Fekete–Szegö type Θ f ( μ ) = a 4 − μ a 2 a 3 and Φ f ( μ ) = a 2 a 4 − μ a 3 2 for a real number μ and for an analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + … , | z | < 1 . This type of research was initiated by Hayami and Owa in 2010. They obtained results for functions satisfying one of the conditions Re f ( z ) / z > α or Re f ′ ( z ) > α , α ∈ [ 0 , 1 ) . Similar estimates were also derived for univalent starlike functions and for univalent convex functions. We discuss Θ f ( μ ) and Φ f ( μ ) for close-to-convex functions such that f ′ ( z ) = h ( z ) / ( 1 − z ) 2 , where h is an analytic function with a positive real part. Many coefficient problems, among others estimating of Θ f ( μ ) , Φ f ( μ ) or the Hankel determinants for close-to-convex functions or univalent functions, are not solved yet. Our results broaden the scope of theoretical results connected with these functionals defined for different subclasses of analytic univalent functions.

scholarly journals Close-to-starlike logharmonic mappings

1996 ◽  
Vol 19 (3) ◽  
pp. 563-573 ◽  
Author(s):  
Zayid Abdulhadi
Keyword(s):  
Unit Disc ◽  
Positive Real Part ◽  
Representation Theorems ◽  
Positive Real ◽  
Distortion Theorems ◽  
Radius Of Univalence

We consider logharmonic mappings of the formf=z|z|2βhg¯defined on the unit discUwhich can be written as the product of a logharmonic mapping with positive real part and a univalent starlike logharmonic mapping. Such mappings will be called close-to-starlike logharmonic mappings. Representation theorems and distortion theorems are obtained. Moreover, we determine the radius of univalence and starlikeness of these mappings.

scholarly journals Properties of Certain Subclass of Meromorphic Multivalent Functions Associated with q-Difference Operator

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1035
Author(s):  
Cai-Mei Yan ◽  
Rekha Srivastava ◽  
Jin-Lin Liu
Keyword(s):  
Difference Operator ◽  
Necessary Conditions ◽  
Partial Sums ◽  
Coefficient Estimates ◽  
Radius Of Starlikeness ◽  
Closure Theorems ◽  
Sufficient And Necessary Conditions ◽  
Distortion Theorems ◽  
Growth And Distortion Theorems

A new subclass Σp,q(α,A,B) of meromorphic multivalent functions is defined by means of a q-difference operator. Some properties of the functions in this new subclass, such as sufficient and necessary conditions, coefficient estimates, growth and distortion theorems, radius of starlikeness and convexity, partial sums and closure theorems, are investigated.

scholarly journals Initial logarithmic coefficients for functions starlike with respect to symmetric points

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Paweł Zaprawa
Keyword(s):  
Exact Result ◽  
Positive Real Part ◽  
Additional Benefit ◽  
Extremal Functions ◽  
Positive Real ◽  
Logarithmic Coefficients ◽  
Symmetric Points

AbstractIn this paper, we obtain the bounds of the initial logarithmic coefficients for functions in the classes $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S of functions which are starlike with respect to symmetric points and convex with respect to symmetric points, respectively. In our research, we use a different approach than the usual one in which the coeffcients of f are expressed by the corresponding coeffcients of functions with positive real part. In what follows, we express the coeffcients of f in $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S by the corresponding coeffcients of Schwarz functions. In the proofs, we apply some inequalities for these functions obtained by Prokhorov and Szynal, by Carlson and by Efraimidis. This approach offers a additional benefit. In many cases, it is easily possible to predict the exact result and to select extremal functions. It is the case for $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S .

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