Bappaditya Bhowmik

On a subclass of meromorphic univalent functions

In this article, we consider a class denoted by which consists of functions f that are holomorphic in the unit disc punctured at a point where f has a simple pole. We prove a sufficient condition for these functions to be univalent in . By using this condition, we construct the family of all functions such that where for some , . Therefore, functions in the class are necessarily univalent. We present some basic properties for functions in the class which include an integral representation formula for such functions and obtain the exact region of variability of the second Taylor coefficient for functions in this class. We also obtain a sharp estimate for the Fekete–Szegö functional defined on the class along with a subordination result for functions in this family. In addition, we obtain some necessary and sufficient coefficient conditions involving the coefficients for functions of the form to be in the class . We have also obtained sharp bounds for , .

On Some Problems of James Miller

Consideramos la clase de funciones univalentes meromoforficos teniendo un polo simple en y la aplicacion del disco unitario sobre el exterior de un dominio el cual es estrellado con respecto al punto . Denotamos esta clase de funciones por . En este articulo encontramos la region exacta de variabilidad del segundo coeficiente de Taylor para funciones in . En vista de estos resultados nosotros rectificamos algunos resultados de James Miller.

Central functions for classes of concave univalent functions

Abstract In this paper we answer a question of Bednarz and Sokól concerning concave univalent functions. We prove that there exist central functions for the classes Co(p) of concave univalent functions with pole at the point z = p ∈ (0, 1). Further, we construct a generalized neighborhood of this central function such that the whole class Co(p) is contained in such neighborhood. We also consider similar questions for the class of functions that are analytic and univalent in the unit disc and for some of its important subclasses.

COEFFICIENT DISCS AND GENERALIZED CENTRAL FUNCTIONS FOR THE CLASS OF CONCAVE SCHLICHT FUNCTIONS

Abstract. We consider functions that map the open unit disc confor-mally onto the complement of an unbounded convex set with openingangle πα, α ∈(1,2], at infinity. We derive the exact interval for the vari-ability of the real Taylor coefficients of these functions and we prove thatthe corresponding complex Taylor coefficients of such functions are con-tained in certain discs lying in the right half plane. In addition, we alsodetermine generalized central functions for the aforesaid class of functions. 1. IntroductionLet A be the class of functions f holomorphic in the unit disc D = {z : |z| <1}, where they have a Taylor expansion at the origin of the following form(1.1) f(z) = z +X ∞n=2 a n (f)z n , z ∈ D.We define Co(α), α ∈ (1,2], as the family of functions f ∈ A such that theysatisfy the following conditions:(i) the functions f ∈ Co(α) are univalent in D,(ii) the set C\ f(D) is convex and the opening angle of f(D) at infinitydoes not exceed πα, α ∈ (1,2],(iii) f(1) = ∞.We call such functions concave univalent functions with opening angle πα atinfinity and we refer to the articles [1, 2, 4, 5] for a detailed discussion on func-tions in this class. We now recall the following characterization for functionsin Co(α) (compare [2, Theorem 2]):