Corrigenda to “Geometric characterization of locally univalent analytic functions, and a generalization”

Corrigenda to “Geometric characterization of locally univalent analytic functions, and a generalization”

by Anatoly Golberg,

Rev. Roumaine Math. Pures Appl. 51 (2006), 5-6 , 633–639

A page is missing in this paper. This is page 639, reproduced below.

Thus, the old page 639 becomes 640.

Letz₀be a point of a domainD, where the functionw=f(z) is continu- ous, and letw₀ =f(z₀). The pointz₀ will be called aU-pointof the mapping w= f(z) if there exist two sequences {z_i},{z_i} (i= 1,2, . . .) of points con- verging to z₀, such that the semitangents t,t at z₀ lie on different straight lines and all the pointsw_i =f(z_i), w_i=f(z_i) are different from w0 =f(z0).

We say that the mappingw =f(z) is orientation preserving at the U- pointz₀ if the sequences{w_i},{w_i}have semitangentsT,T atw₀ such that if 0<{t,t}< πthen 0≤ {T,T}< π.

Trokhimchuk [8] gave the generalization below of the Bohr and Menshoff theorems for arbitrary continuous functions.

Theorem F. Let w = f(z) be a continuous mapping of the domain D with constant stretching at almost every point z in D, and let this mapping be orientation preserving at almost every U-point. Then the function f(z) is analytic in D. In addition, if there are no U-points of f in D, then f(z) ≡ const.

6. GEOMETRIC CONDITIONS FOR ANALYTICITY OF CONTINUOUS FUNCTIONS

Using the local geometric characteristics for arbitrary continuous func- tionsw=f(z) at a pointz∈Dgiven in Section 3, for 1≤p <∞ we denote byI_p(z, t) and O_p(z, t) the quantities

(7) I_p(z, t) = R^∗(z, t) r(z, t)

R(z, t) r^∗(z, t)

_p−1 ,

684 Corrigenda 2

(8) O_p(z, t) =

R^∗(z, t) r(z, t)

_p−1

R(z, t) r^∗(z, t).

Since in the case of continuous functions the characteristic neighborhood G_t(z) can be shrunk to a point or to an arc, the quantities (7) and (8) are not determined in the general case except for univalent functions. But, for analytic functions, the values I_p(z, t) and O_p(z, t) are finite at least for sufficiently smallt.

Theorem 2. Let a functionf(z) be continuous in a domain D. Assume that for almost every point z ∈ D there exists a normal regular system of neighborhoods {Gt(z)} ⊂ D such that the quantities Ip(z, t) and Op(z, t) are finite for sufficiently smallt >0, and that f is orientation preserving at every U-point. If either the inequality

lim sup

t→0 I_p(z, t)≤

Θ(z)^2−p

2