Krzysztof Piejko

On the Dziok–Srivastava operator under multivalent analytic functions

Abstract The aim of this paper is to investigate various properties and characteristics of the Dziok–Srivastava operator introduced in Dziok and Srivastava [J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999) 1–13]. Our paper is motivated essentially by the familiar work Liu and Srivastava [J.-L. Liu, H.M. Srivastava, Certain properties of the Dziok–Srivastava operator, Appl. Math. Comput. 159 (2004) 485–493] which has been recently published.

A note involving p-valently Bazilević functions

A theorem involving p-valently Bazilevic functions is considered and then its certain consequences are given.

On some Problem of the Convolution of Bounded Functions

Abstract We consider convolution properties of regular functions subordinated to the homography 1 + A z 1 − B z . We give some extension of result from [11].

Convolution of functions with free normalization

Let A denotes the unit disc and let Ti. be the class of functions regular in A. For two functions f(z) = anzn, g(z) — bnzn of the class H we define the convolution as follows ( / * g)(z) := anbnzn. In this paper we determine the domain D3 such that (f + g) (A) C D3 for every two functions f,g € H said such that / (A) C D\, g{A) C D2, where D\ and £>2 are the given circular (disc, halfplane) or angularly domains.

Hadamard product of analytic functions and some special regions and curves

In this paper we present some new applications of convolution and subordination in geometric function theory. The paper deals with several ideas and techniques used in this topic. Besides being an application to those results, it provides interesting corollaries concerning special functions, regions and curves.MSC:30C45, 30C80.

On

We define in this paper new distance generalizations of the Pell numbers and the companion Pell numbers. We give a graph interpretation of these numbers with respect to a special 3-edge colouring of the graph.

k

Abstract In this work we present some new results on convolution and subordination in geometric function theory. We prove that the class of convex functions of order α is closed under convolution with a prestarlike function of the same order. Using this, we prove that subordination under the convex function order α is preserved under convolution with a prestarlike function of the same order. Moreover, we find a subordinating factor sequence for the class of convex functions. The work deals with several ideas and techniques used in geometric function theory, contained in the book Convolutions in Geometric Function Theory by Ruscheweyh (1982).

-Distance Pell Numbers in 3-Edge-Coloured Graphs

In this paper we consider convolution properties of a class of bounded analytic functions investigated by J. Stankiewicz and Z. Stankiewicz in [6], We give some examples which verify a conjecture connected with this paper.

On the convolution and subordination of convex functions

Abstract In [RUSCHEWEYH, S.-SHEIL-SMALL, T.: Hadamard product of schlicht functions and the Poyla-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119-135] the authors proved the P`olya-Schoenberg conjecture that the class of convex univalent functions is preserved under convolution, namely K ∗ K = K. They proved also that the class of starlike functions and the class of close-to-convex functions are closed under convolution with the class K. In this paper we consider similar convolution problems for some classes of functions. Especially we give a new applications of a result [SOKÓŁ, J.: Convolution and subordination in the convex hull of convex mappings, Appl. Math. Lett. 19 (2006), 303-306] on the subordinating relations in the convex hull of convex mappings under convolution. The paper deals with several ideas and techniques used in geometric function theory. Besides being an application to those results it provides interesting corollaries concerning special functions, regions and curves.

Convolution properties of a class of analytic functions

We determine the smallest and the largest number of -edge colourings in trees. We prove that the star is a unique tree that maximizes the number of all of the -edge colourings and that the path is a unique tree that minimizes it.