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A note on certain fractional operator. (English) Zbl 0754.30009

Let S denote the class of functions f(z)=z+ n=2 a n z n which are analytic and univalent in the unit disk U={z|z|<1}. Let n be a non-negative integer, and let α be such that 0α<1. For f(z)S, the authors consider D z n+α f(z), the fractional derivative of order n+α, and conjecture that

|D z n+α f(z)|Γ(n+1+α)(n+α+|z|) (1-|z|) n+2+α

for zU, with equality when f(z) is the Koebe function k(z)=z (1-z) 2 (zU). If true, this will generalize an earlier result of Landau (1926), namely:

|f (n) (z)|n!(n+|z|) (1-|z|) n+2

for zU, n=1,2,3,, and f(z)S.

30C45Special classes of univalent and multivalent functions
26A24Differentiation of functions of one real variable