TAILIEUCHUNG - Some properties of a new integral operator

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. | JOURNAL OF SCIENCE OF HNUE DOI Mathematical and Physical Sci. 2015 Vol. 60 No. 7 pp. 3-14 This paper is available online at http 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 lt 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 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 lt 1 if and only if 1 zf z 1 zf z Re 1 gt λ or lt 1 λ z U . 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 lt 1 and we denote this class by S β if it satisfies zf z zf z Re gt β or lt β z U. f z f z

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