Differential Subordination and Superordination For Analytic Functions Defined Using A Family Of Generalized Differential Operators
نویسنده
چکیده
By making use of the generalized differential operator, the authors derive the subordination and superordination results for certain normalized analytic functions in the open unit disk. Many of the well-known and new results are shown to follow as special cases of our results. 1 Preliminaries Let H be the class of functions analytic in the open unit disc U = {z : | z |< 1}. Let H(a, n) be the subclass of H consisting of functions of the form f(z) = a+ anz n + an+1z n+1 + . . .. Let An = {f ∈ H, f(z) = z + an+1z n+1 + an+2z n+2 + . . .} and let A = A1. Let the functions f and g be analytic in U . We say that the function f is subordinate to g if there exists a Schwarz function w, analytic in U with w(0) = 0 and | w(z) |< 1 such that f(z) = g(w(z)) for z ∈ U . We denote it by f(z) ≺ g(z). In particular, if the function g is univalent in U , the above subordination is equivalent to f(0) = g(0) and f(U) ⊂ g(U). Let p, h ∈ H
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