In this paper, some theorems involving inequalities on p-valent functions (that is p-valenty close-to-convex functions, p-valently starlike functions, and p-valently convex functions) are given. Moreover, some applications in the theorems which are important for geometric function theory are also included. | Turk J Math 23 (1999) , 453 – 459. ¨ ITAK ˙ c TUB SOME THEOREMS INVOLVING INEQUALITIES ON P-VALENT FUNCTIONS ¨ H¨ useyin Irmak, Omer Faruk C ¸ etin Abstract In this paper, some theorems involving inequalities on p-valent functions (that is p-valenty close-to-convex functions, p-valently starlike functions, and p-valently convex functions) are given. Moreover, some applications in the theorems which are important for geometric function theory are also included. Keywords: Analytic, p-valent, p-valently colse-to-convex functions, p-valently starlike functions, p-valently convex functions, open unit disk, and Jack’s Lemma. 1. Introduction and Definitions Let T (p) denote the class of functions f(z) of the form f(z) = z p + ∞ X ak z k , (p ∈ N = {1, 2, 3, . . .}), (1) k=p+1 which are analytic and p-valent in the open unit disk U = {z : z ∈ C and | z | 0, (z ∈ U ; p ∈ N ). (2) On the other hand, a function f(z) ∈ T (p) is said to be in the subclass T S(p) of p-valently starlike functions with respect to the origin in U if it satisfies the inequality 453 ˙ IRMAK, C ¸ ETIN (cf.[1-3]): Re zf 0 (z) f(z) > 0, (z ∈ U ; p ∈ N ). (3) Furthermore, a function f(z) ∈ T (p) is said to be in the subclass T C(p) of p-valently convex functions with respect to the origin in U if it satisfies the inequality (cf.[1-3]): zf 00 (z) > 0, (z ∈ U ; p ∈ N ). Re 1 + 0 f (z) (4) A function f(z) ∈ T (p) is said to be in the subclass Vq (p) if it satisfies the inequality: (p − q)! Dzq f(z) q; p ∈ N ; q ∈ N0 = N ∪ {0}), and, a function f(z) ∈ T (p) is said to be in the subclass Wq (p) if it satisfies the inequality: q+1 zDz f(z) q; p ∈ N ; q ∈ N0 ). Here and throughout this paper, Dzq denotes the q th-order ordinary differential operator. For a function f(z) ∈ T (p), Dzq f(z) ∞ X p! k! p−q z ak z k−q , (p > q; p ∈ N ; q ∈ N0 ). = + (p − q)! (k − q)! (7) k=p+1 To establish our results, we need the following Lemma given by Jack [4] (also, by Miller
