SANDWICH-TYPE RESULTS FOR A CLASS OF CONVEX INTEGRAL OPERATORS

Let H(U) be the space of analytic functions in the unit disk U. For the integral operator A(alpha,beta,gamma)(phi,phi) : K -> H(U), with K subset of H(U), defined by A(alpha,beta,gamma)(phi,phi)[f](z) = [beta + gamma/z(gamma)phi(z) integral(z)(0) f(alpha)(t)phi(t)t(delta-1)dt](1/beta), where alpha, beta, gamma, delta is an element of C and phi, phi is an element of H(U), we will determine sufficient conditions on g(1), g(2), alpha, beta and gamma, such that z phi(z)[g(1)(z)/z](alpha) (sic) z phi(z)[f(z)/z](alpha) (sic) z phi(z)[g(2)(z)/z](alpha) implies z phi(z)[A(alpha,beta,gamma)(phi,phi)[f](z)/z](beta) (sic) z phi(z)[A(alpha,)beta,gamma(phi,phi)[g(2)](z)/z](beta). The symbol "(sic)" stands for subordination, and we call such a kind of result a sandwich-type theorem. In addition, z phi(z) [A(alpha,beta,gamma)(phi,phi)[g(1)](z)/z](beta) is the largest function and z phi(z) [A(alpha,beta,gamma)(phi,phi)[g(2)](z)/z](beta) the smallest function so that the left-hand side, respectively the right-hand side of the above implications hold, for all f functions satisfying the assumption. We give a particular case of the main result obtained for appropriate choices of functions phi and phi, that also generalizes classic results of the theory of differential subordination and superordination.