AN INTEGRAL UNIVALENT OPERATOR OF THE CLASS S ( p ) AND T 2

Let A be a class of the analytic functions { f : f(z) = z + a2z + ... } , z ∈ U , where U is the unit disk, U = {z : |z| < 1} . By S we denote the subclass of A consisting of function univalent in the unit disk. Let p be a real number with the property 0 < p ≤ 2. We define the class S (p) as the class of the functions f ∈ A that satisfy the conditions f (z) 6= 0 and ∣(z/f (z))′′ ∣∣ ≤ p, z ∈ U. Also, if f ∈ S (p) then the following property is true ∣∣ z2f ′(z) f2(z) − 1 ∣∣ ≤ p |z| , z ∈ U . This relation was proved in [4]. We denote by T2 a class of the univalent functions that satisfy the condition ∣∣ z2f ′(z) f2(z) − 1 ∣∣ < 1, z ∈ U , and also have the property f ′′ (0) = 0. These functions have the form f(z) = z+a3z+a4z+.... For 0 < μ < 1 we define a subclass of T2 containing the functions that satisfy the property ∣∣ z2f ′(z) f2(z) − 1 ∣∣ < μ < 1, z ∈ U . We denote that class by Tμ,2.