Some properties of a new integral operator

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For certain classes of analytic functions fi, gi and hi, i=1,..., n in the open unit disk U, we define a new general integral operator Hn(z) = R z 0 Qn i=1 fi(t) hi(t)αi (g'i (t))yi dt and we study convexity properties of this integral operator. Keywords: Analytic functions, integral operators, convexity order, convex funtions, order.

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Nội dung Text: Some properties of a new integral operator

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0025 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 3-14 This paper is available online at http://stdb.hnue.edu.vn SOME PROPERTIES OF A NEW INTEGRAL OPERATOR Nguyen Van Tuan1 , Adriana Oprea1 and Daniel Breaz2 1 Department of Mathematics, University of Pitesti, Romania 21 Decembrie, University of Alba Iulia, Romania Abstract. For certain classes of analytic functions fi , gi and hi , i = 1, . . . , n in the open unit disk U , we define a new general integral operator Hn (z) = R z Qn fi (t) αi ′ γ 0 i=1 hi (t) (gi (t)) i dt and we study convexity properties of this integral operator. Keywords: Analytic functions, integral operators, convexity order, convex funtions, order. 1. Introduction Let U = {z : |z| < 1} be the unit disk and A be the class of all functions in the form ∞ X f (z) = z + ak z k , z ∈ U (1.1) k=2 which are analytic in U and satisfy the conditions f (0) = f ′ (0) − 1 = 0. We denote by S the subclass of A consisting of univalent functions on U . In [1] and [2] they were defined the following classes of functions. A function f ∈ A is a convex function of complex order b, (b ∈ C \ {0}) and type λ (0 ≤ λ < 1), if and only if
1 zf ′′ (z)
1 zf ′′ (z)
Re 1 + > λ or
< 1 − λ, (z ∈ U ). (1.2) b f ′ (z)
b f ′ (z)
We denote by Cλ∗ (b) the class of these functions. A function f ∈ A is a starlike function of order β, 0 ≤ β < 1 and we denote this class by ∗ S (β) if it satisfies ′
′
zf (z)
zf (z)
Re > β or
< β, z ∈ U. (1.3) f (z)
f (z)