Bol. Soc. Paran. Mat. (3s.) v. 33 2 (2015): 217–230. c SPM –ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v33i2.24028 Majorization Problems and Integral Transforms for a Class of Univalent Functions with Missing Coefficients Som P. Goyal, Rakesh Kumar and Teodor Bulboacă abstract: In 2005, Ponnusamy and Sahoo have introduced a special subclass of univalent functions Un(λ) (n ∈ N, λ> 0) and obtained some geometrical proper- ties, including strongly starlikeness and convexity, for the functions of this subclass Un(λ). Moreover, they have studied some important properties of an integral trans- form connected with these subclasses. The aim of the present paper is to investigate another important concept of majorization for the functions belonging to the class Un(λ) (0 <λ ≤ 1). We shall also discuss a majorization problem for some special integral transforms. Key Words: Univalent functions, quasi-subordination, starlike functions, ma- jorization property, integral transforms. Contents 1 Introduction and Preliminaries 217 2 Majorization problem for the class U n (λ) 220 3 Integral Transforms 223 1. Introduction and Preliminaries Let H denote the class of functions which are analytic in the open unit disc Δ= {z ∈ C : |z | < 1}. For a fixed n ∈ N = {1, 2,... }, let A n be the class of functions f ∈ H of the form f (z )= z + ∞  k=n+1 a k z k ,z ∈ Δ. (1.1) We denote A := A 1 , while the subclass of A consisting of all univalent functions in Δ is denoted by S. Definition 1.1. [11, p. 226] If f,g ∈ H, then f is said to be subordinate to g, if there exists a function w ∈ H satisfying w(0) = 0 and |w(z )| < 1, z ∈ Δ, such that f (z )= g(w(z )),z ∈ Δ. The subordination relation is denoted by f (z ) ≺ g(z ). (1.2) 2000 Mathematics Subject Classification: 30C45, 30C80. 217 Typeset by B S P M style. c  Soc. Paran. de Mat.