A General Class of Integral Operators Preserving Subordinations and Superordinations
by
Teodor Bulboacă
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania

If H(U) denotes the space of analytic functions in the unit disk U, for the integral operator A_{a, b, g}^h:K→ H(U), with K ⊂ H(U), defined by

Aa, b, gh[f](z)= é ë b+g zg ó õ z 0  fa(t)h(t)td-1dt ù û 1/b   ,

where a, b, g, d ∈ C and h ∈ H(U), we determined sufficient conditions on g₁, g₂, a, b and g such that

zh(z)[g1(z)/z]a << zh(z)[f(z)/z]a << zh(z)[g2(z)/z]a

implies

z[Aa, b, gh[g1](z)/z]b << z[Aa, b, gh[f](z)/z]b << z[Aa, b, gh[g2](z)/z]b,

where the symbol " << " represents the subordination.

In addition, both of the subordinations are sharp, since z[A_{a, b, g}^h[g₁](z)/z]^b is the largest function and z[A_{a, b, g}^h[g₂](z)/z]^b is the smallest function so that the left-hand side, respectively the right-hand side of the above implication hold, for all f functions satisfying the differential subordination, respectively the differential superordination of the assumption. The results generalizes those of the last author, obtained for the special case a = b and h ≡ 1.

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Date received: January 31, 2010