Janusz Sokół

Coefficient Estimates in a Class of Strongly Starlike Functions

In this paper we consider some coe-cient estimates in the subclass SL ⁄ of strongly starlike functions deflned by a certain geometric condition.

On a subclass of strongly starlike functions

Abstract Let S ∗ ( q c ) , c ∈ ( 0 , 1 ] , denote the class of analytic functions f in the unit disc U normalized by f ( 0 ) = f ′ ( 0 ) − 1 = 0 and satisfying the condition | [ z f ′ ( z ) / f ( z ) ] 2 − 1 | ∣ c , z ∈ U . The relations between S ∗ ( q c ) and other classes geometrically defined are considered. The radii of convexity (starlikeness) of order α are calculated. The same problem in the class of strongly starlike functions of order β is also considered.

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.

On some applications of the Dziok-Srivastava operator

Abstract Carlson and Shaffer [B.C. Carlson, D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984) 737–745] have introduced a linear operator associated with the Gaussian hypergeometric function which has been generalized by 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]. Certain classes of analytic functions defined by means of those operators have been considered in [J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform. Spec. Funct. 14 (2003) 7–18; J. Dziok, H.M. Srivastava, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv. Stud. Contemp. Math. 5 (2) (2002) 115–125] and recently in [J. Dziok, On some applications of the Briot–Bouquet differential subordination, J. Math. Anal. Appl. 328 (2007) 295–301; J. Dziok, Some relations including various linear operators, Demonstratio Math. XL (2007) 77–84; J.-L. Liu, H.M. Srivastava, Certain properties of the Dziok–Srivastava operator, Appl. Math. Comput. 159 (2004) 485–493]. In the present paper, new results for a familiar class of multivalent functions have been obtained. We have used the methods of differential subordination and the properties of Hadamard product.

On sufficient condition to be in a certain subclass of starlike functions defined by subordination

Abstract An interesting criterion was given by Tuneski [N. Tuneski, On the quotient of the representations of convexity and starlikeness, Math. Nachr. 248–249 (2003) 200–203] for the analytic functions to be in the class S ∗ [ 1 + Az 1 + Bz ] and its subclasses, where - 1 ⩽ B A ⩽ 1 . This result is an extension of an earlier result of Silverman [H. Silverman, Convex and starlike criteria, Int. Math. Math. Sci. 22 (1) (1999) 75–79] for α-starlike functions. In this paper we give a generalization of main theorem contained in Tuneski (2003). Some applications involving this result are also considered.

A certain class of starlike functions

This paper presents a new class of functions analytic in the open unit disc, and closely related to the class of starlike functions. Besides being an introduction to this field, it provides an interesting connections defined class with well known classes. The paper deals with several ideas and techniques used in geometric function theory. The order of starlikeness in the class of convex functions of negative order is also considered here.

ON CERTAIN CLASS OF MEROMORPHIC FUNCTIONS WITH POSITIVE COEFFICIENTS

Abstract In the present investigation we define a new class of meromorphic functions on the punctured unit disk Δ * : = { z ∈ ℂ : 0 | z | 1 } by making use of the generalized Dziok–Srivastava operator H m l [ α 1 ] . Coefficient inequalities, growth and distortion inequalities, as well as closure results are obtained. We also establish some results concerning the partial sums of meromorphic functions and neighborhood results for functions in new class.

On certain problem in the class of k-starlike functions

In this paper we consider the classes of k-uniformly convex and k-starlike functions defined in Kanas and Wisniowska (1999, 2000) [1,2] which generalize the class of uniformly convex functions introduced by Goodman (1991) [3]. We discuss the real part of f(z)/z, when f is k-starlike. We find the minimum of Ref(z)/z improving the results obtained recently in Wisniowska-Wajnryb (2009) [11].

On α-convex functions related to shell-like functions connected with Fibonacci numbers

Abstract This paper presents a new class SLM α of functions f(z) analytic and normalized in the open unit disc U = { z : | z | 1 } (which is related to a shell-like curve and associated with Fibonacci numbers) satisfying the condition that α 1 + zf ″ ( z ) f ′ ( z ) + ( 1 - α ) zf ′ ( z ) f ( z ) ∈ p ˜ ( U ) ( z ∈ U ) , where α is a real number and p ˜ ( z ) = τ z + τ 2 z 2 1 - τ z - τ 2 z 2 ( τ = ( 1 - 5 ) / 2 ; z ∈ U ) . The class SLM α being closely related to the classes of starlike and convex functions, we apply some basic techniques to investigate certain interesting properties (given below) for this class of functions. Some important observations of the main results are also mentioned.

On some sufficient conditions for univalence and starlikeness

In this work, the conditions for univalence, starlikeness and convexity are discussed.MSC:30C45, 30C80.