LIU, Modified Roper-Suffridge operator for some

holomorphic mappings, Front.

Skrypnik, "On

holomorphic solutions of the Darwin equations of motion of point charges," Ukrainian Mathematical Journal, vol.

(Idea of the proof) by the standard

holomorphic coordinate changes, r(w) has the Taylor series expansion as in (8).

We denote the Smirnov class by [N.sub.*](U), which consists of all

holomorphic functions f on U such that log(1 + [absolute value of (f(z))]) [less than or equal to] Q[[phi]](z) (z [member of] U) for some [phi] [member of] [L.sup.1](T), [phi] [greater than or equal to] 0, where the right side denotes the Poisson integral of [phi] on U.

Holland and Walsh [7] characterized

holomorphic Bloch space in D in terms of weighted Euclidian Lipschitz functions of indices (1/2,1/2).

in [epsilon] with coefficients [a.sub.i](z) in the ring O(r) of

holomorphic functions on [D.sub.r], continuous in its closure, satisfying

Shafikov, "Analytic continuation of

holomorphic mappings from nonminimal hypersurfaces," Indiana University Mathematics Journal, vol.

where a tangent space index A = 1, ..., 6 has been split into a

holomorphic index i = 1,2,3 and an antiholomorphic index [bar.i] = 1,2,3.

If f: D \ E c O is

holomorphic and bounded, then f has a unique

holomorphic extension to D.

Consider the

holomorphic function F(z) := [??](z) - z defined on [B.sub.[delta]].

If P (or F) is parallel, then the

holomorphic distribution H is intergrable.

On [R.sup.2], a 1-quasiconformal mapping is

holomorphic or antiholomorphic.

If D is a non-empty simply connected open subset of the complex plane C which is not all of C, then there exists a biholomorphic (bijective and

holomorphic) mapping f from D onto the open unit disk U = {z [member of] C :[absolute value of z] <1} (Krantz, 1999, Section 6.4.3, p.

Let [A.sup.*.sub.n[zeta]] = {f [member of] H(U x [bar.U]), f(z, [zeta]) = z + [a.sub.n+1]([zeta])[z.sup.n+1] + ***, z [member of] U, [zeta] [member of] [bar.U]}, with [A.sup.*.sub.1[zeta]] = [A.sup.*.sub.[zeta]], where [a.sub.k]([zeta]) are

holomorphic functions in [bar.U] for k [greater than or equal to] 2, and

with

holomorphic coefficients [a.sub.k] (t, z) on some domain D [subset] C with respect to t and near the origin in C with respect to z.