ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2013, Vol. 48, No. 6, pp. 297–309. c  Allerton Press, Inc., 2013. Original Russian Text c  M. K. Aouf, T. M. Seoudy, 2013, published in Izvestiya NAN Armenii. Matematika, 2013, No. 6, pp. 3-14. REAL AND COMPLEX ANALYSIS Some Properties of Certain Classes of p−valent Functions Defined by the Hadamard Product M. K. Aouf 1* and T. M. Seoudy 2** 1 Mansoura University, Mansoura, Egypt 2 Fayoum University, Fayoum, Egypt Received September 5, 2012 Abstract—In this paper we obtain sandwich type theorems, inclusion relationships, convolution properties and coefficient estimates of certain classes of p−valent analytic functions defined by a convolution. Several other new results are also obtained. MSC2010 numbers : 30C45 DOI: 10.3103/S106836231306006X Keywords: p−valent functions; subordination; superordination; linear operator; Hadamard product; convolution. 1. INTRODUCTION Let H be the class of functions analytic in the open unit disk U = {z ∈ C : |z| < 1} and let H [a, n] be the subclass of H consisting of function of the form: f (z)= a + a n z n + a n+1 z n+1 + ... (z ∈ U ) . (1.1) Let A (p) denote the class of all analytic functions of the form: f (z)= z p + ∞  k=1 a p+k z p+k (p ∈ N = {1, 2, 3, ...} ; z ∈ U ). (1.2) We set A (1) = A. If f (z) and g (z) are analytic in U functions, we say that f (z) is subordinate to g (z), or equivalently, g (z) is superordinate to f (z) , written symbolically f (z) ≺ g(z)(z ∈ U ) , if there exists a Schwarz function ω (z), which (by definition) is analytic in U with ω (0) = 0 and |ω (z)| < 1 such that f (z)= g(ω(z)) (z ∈ U ). Indeed, it is known that f (z) ≺ g(z)= ⇒ f (0) = g(0) and f (U ) ⊂ g(U ). Furthermore, if the function g (z) is univalent in U , then we have the following equivalence (see [5] , [18] and [19]): f (z) ≺ g(z) ⇐⇒ f (0) = g(0) and f (U ) ⊂ g(U ). For functions f (z) given by (1.2) and g (z)= z p + ∞  k=1 b p+k z p+k (p ∈ N; z ∈ U ) , (1.3) the Hadamard product or convolution of f (z) and g (z) is defined by (f ∗ g)(z)= z p + ∞  k=1 a p+k b p+k z p+k =(g ∗ f )(z) . * E-mail: mkaouf127@yahoo.com ** E-mail: tms00@fayoum.edu.eg 297