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A class of analytic functions defined by Ruscheweyh derivative

K. S. Padmanabhan, M. Jayamala (1991)

Annales Polonici Mathematici

The function f ( z ) = z p + k = 1 a p + k z p + k (p ∈ ℕ = 1,2,3,...) analytic in the unit disk E is said to be in the class K n , p ( h ) if ( D n + p f ) / ( D n + p - 1 f ) h , where D n + p - 1 f = ( z p ) / ( ( 1 - z ) p + n ) * f and h is convex univalent in E with h(0) = 1. We study the class K n , p ( h ) and investigate whether the inclusion relation K n + 1 , p ( h ) K n , p ( h ) holds for p > 1. Some coefficient estimates for the class are also obtained. The class A n , p ( a , h ) of functions satisfying the condition a * ( D n + p f ) / ( D n + p - 1 f ) + ( 1 - a ) * ( D n + p + 1 f ) / ( D n + p f ) h is also studied.

A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions

Karl-Joachim Wirths (2004)

Annales Polonici Mathematici

Let D denote the open unit disc and f:D → ℂ̅ be meromorphic and injective in D. We further assume that f has a simple pole at the point p ∈ (0,1) and an expansion f ( z ) = z + n = 2 a ( f ) z , |z| < p. In particular, we consider f that map D onto a domain whose complement with respect to ℂ̅ is convex. Because of the shape of f(D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p). It is proved that for p ∈ (0,1) the domain of variability of the coefficient...

A subclass of harmonic functions with varying arguments defined by Dziok-Srivastava operator

G. Murugusundaramoorthy, Kaliappan Vijaya, Ravinder Krishna Raina (2009)

Archivum Mathematicum

Making use of the Dziok-Srivastava operator, we introduce a new class of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc and are related to uniformly convex functions. We investigate the coefficient bounds, distortion inequalities and extreme points for this generalized class of functions.

An extension of typically-real functions and associated orthogonal polynomials

Iwona Naraniecka, Jan Szynal, Anna Tatarczak (2011)

Annales UMCS, Mathematica

Two-parameters extension of the family of typically-real functions is studied. The definition is obtained by the Stjeltjes integral formula. The kernel function in this definition serves as a generating function for some family of orthogonal polynomials generalizing Chebyshev polynomials of the second kind. The results of this paper concern the exact region of local univalence, bounds for the radius of univalence, the coefficient problems within the considered family as well as the basic properties...

Application of Salagean and Ruscheweyh Operators on Univalent Holomorphic Functions with Finitely many Coefficients

Najafzadeh, Shahram (2010)

Fractional Calculus and Applied Analysis

MSC 2010: 30C45, 30C50The purpose of the present paper is to introduce a new subclass of holomorphic univalent functions with negative and fixed finitely coefficient based on Salagean and Ruscheweyh differential operators. The various results investigated in this paper include coefficient estimates, extreme points and Radii properties.

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