Abstract

Hacettepe Journal of Mathematics and Statistics
Volume 40 (6) (2011), 829 – 837
SOME CONVEXITY PROPERTIES FOR TWO
NEW P -VALENT INTEGRAL OPERATORS
Erhan Deniz∗†, Murat Çağlar∗ and Halit Orhan∗
Received 11 : 11 : 2010 : Accepted 09 : 05 : 2011
Abstract
In this paper, we define two new general p-valent integral operators in
the unit disc U, and obtain the convexity properties of these integral
operators of p-valent functions on some classes of β-uniformly p-valent
starlike and β-uniformly p-valent convex functions of complexorder.
µ As
Rz
special cases, the convexity properties of the operators 0 f (t)
dt
t
Rz ′
µ
and 0 (g (t)) dt are given.
Keywords: Analytic functions, Integral operators, β-uniformly p-valent starlike and
β-uniformly p-valent convex functions, Complex order.
2000 AMS Classification: Primary 30 C 80. Secondary 30 C 45.
1. Introduction and preliminaries
Let Ap denote the class of functions of the form
(1.1)
f (z) = z p +
∞
X
ak z k , (p ∈ N = {1, 2, . . . , }) ,
k=p+1
which are analytic in the open disc U = {z ∈ C : |z| < 1}.
A function f ∈ S∗p (γ, α) is p−valently starlike of complex order γ (γ ∈ C − {0}) and
type α (0 ≤ α < p), that is, f ∈ S∗p (γ, α), if it satisfies the following inequality;
1 zf ′ (z)
(1.2)
ℜ p+
−p
> α, (z ∈ U) .
γ
f (z)
∗Department of Mathematics, Faculty of Science, Atatürk University, TR-25240 Erzurum, Turkey. E-mail: (E. Deniz) [email protected] (M. Çağlar) [email protected]
(H. Orhan) [email protected]
†
Corresponding Author.