Vol. 79, No. 1, 1978

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Starlike integral operators

Sanford S. Miller, Petru T. Mocanu and Maxwell O. Reade

Vol. 79 (1978), No. 1, 157–168
Abstract

Let S denote the set of all functions f(z) = z + univalent and starlike in the unit disc U. The authors show that integral operators of the form

              ∫
-β-+-γ   z α      δ−1  1∕β
ℐ(f) ≡ [zγΦ (z)  0 f (t)φ(t)t   dt]  = z + ⋅⋅⋅

with suitable restrictions on the analytic functions Φ(z) and φ(z), for suitable choices of the constants α, β, γ, and δ, transforms S into S. With other restrictions on the parameters, the authors obtain transformations of the sets K, S×K, K ×K into S. Here K denotes, as usual, the set of all f(z) = z + univalent and convex in U.

Mathematical Subject Classification 2000
Primary: 30C45
Milestones
Received: 21 January 1978
Published: 1 November 1978
Authors
Sanford S. Miller
Petru T. Mocanu
Maxwell O. Reade