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Koebe domains for the classes of functions with ranges included in given sets

100%

Koczan L. , Zaprawa P.

2008

tom Vol. 14, nr 1

43-52

In this paper we present a new method of determining Koebe domains. We apple this method by giving a new proof of the well-known theorem of A. W. Goodman concerning the Koebe domain for the class T of typically real functions. We applied also the method to determine Koebe sets for classes of the special type , i.e. for TM,g = {∫ ∈ T : ∫(Δ) ⊂ Mg(Δ)}, g ∈ T ∩ S, M > 1, where Δ = {z ∈ C: IzI < 1} and T, S stand for the classes of tipically real functions and univalent functions respectively. In particular, we find the Koebe domains for the class T (M) of all typically real functions with ranges in a given strip.

The class of functions defined by differential integral operators

100%

Trojnar-Spelina L.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

1999

tom z. 23 [175]

145-160

In this paper we consider the class of functions with negative coefficients denned by differential integral operators. Among others the necessary and sufficient condition for a function f to be in the studied class is presented.

The classes of functions which can be defined by subordination

86%

Jurasińska R. , Szpila A.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

1998

tom z. 22 [170]

41-49

Let H = H(U) be the class of all functions which are holomorphic in the unit disc U = {z : \z\ < 1}. Let P(n) denotes the class of all functions p(z) = 1+piz+... is an element H, such thatp(pz) -< (1+zn/(1-zn), where -< denotes subordination. With the class P(n) we connect the subclass S*(n) of starlike functions in the following way. A function f(z) = z + a2z + ... belongs to S* (n) if and only if zf'(z)/f(z) is an element of P(n). In this note we give the estimations of some coefficients in the classes P(n) and S*(n) and we find the radius of convexity of the class S*(n).

On certain differential subordination for circular domains

86%

Lecko A.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

1998

tom z. 22 [170]

101-108

Let U = {z : \z\ < 1} denote the unit disk. For A and B such that -1

The convexity of Hadamard product of three functions

72%

Sokół J.

2007

tom Vol. 29

121-125

Let α, β, γ < and let f, g, h be analytic functions such that Re[f'(z)] ≥ α, Re[g'(z)] ≥ β, Re[h'(z)] ≥ γ in the unit disc U. In this paper we give a sufficient condition for the convexity of Hadamard product f *g *h in the unit dsc U.

Certain differential subordinations using a generalized Salagean operator and Ruscheweyh operator

72%

Lupas A. A.

2010

tom Vol. 33

67-72

In the present paper we define a new operator using the generalized Salagean operator and Ruscheweyh operator. Denote by [formula/wzór] the Hadamard product of the generalized Salagean operator [formula/wzór] and Ruscheweyh operator R(n), given by [fomula/wzór] is the class of normalized analytic functions with A1 = A. We study some differential subordinations regarding the operator [formula/wzór].

Generalized classes of uniformly convex functions

72%

Dziok J. , Szpila A.

2010

tom Vol. 33

13-26

In this paper we introduce some subclasses of analytic functions with varying argument of coeffcients. These classes are defined in terms of the Hadamard product and generalize the well-known classes of uniformly convex functions. We investigate the coeffcients estimates, distortion properties, radii of starlikeness and convexity for defined classes of functions.

On classes of functions defined by subordination to some special n-valent functions

72%

Jurasińska R.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

1999

tom z. 23 [175]

52-60

Let H = H(U) be the class of all functions which are holomorphic in the unit disc U = {z : \z[ < 1}. Let P(n,A,B) denotes the class of all functions p(z) = 1 +p1z +p2z2 + ...is an element of H, such that p(z) -< 1+Azn/1-Bzn, where -< denotes subordination. With the class P(n, A, B) we connect the subclass S*(n, A, B) of starlike functions in the following way. A function f(z) = z o+a2z.2 z2 + ... belongs to S*(n, A, B) if and only ifzf'(z)/f(z) is an element of P(n, A, B). In this note we give some estimations for the modulus of functions and coefficients in the classes P(n,A,B) and S*(n,A, B).

On some subclass of strongly starlike functions

72%

Sokół J.

1998

tom Vol. 31, nr 1

81-86