Some characteristic properties of analytic functions

Abstract

In this paper, we consider a class L(λ, μ; ϕ) of analytic functions f defined in the open unit disk U satisfying the subordination condition that,where is the Sălăgean operator and ϕ(z) is a convex function with positive real part in U. We obtain some characteristic properties giving the coefficient inequality, radius and subordination results, and an inclusion result for the above class when the function ϕ(z) is a bilinear mapping in the open unit disk. For these functions f (z) ; sharp bounds for the initial coefficient and for the Fekete-Szegö functional are determined, and also some integral representations are given.

Some Geometric Properties of Analytic Functions Involving a New Fractional Operator

“…We note that the fractional operator D ν,0 0 defined by (1.2) is precisely the Ruscheweyh derivative operator R ν of order ν (ν > −1) and D 0,0 λ is the fractional differintegral operator Ω λ z of order λ (−∞ < λ < 2) , while D 0,n 0 = D n and D 1−λ,n λ = D n+1 are the Sȃlȃgean operators, respectively, of order n and n + 1 (n ∈ N 0 ) . There are numerous results in the literature on the Geometric Function Theory which are based on the use of the Ruscheweyh, Sȃlȃgean, and the fractional differintegral operators (see, for instance, the works in [6][7][8][9][10]13,17,[19][20][21][22][23]26,28,29]; see also [27] (and the references cited therein).…”