To resolve the Riemann-Hilbert problem in the case of domains bounded by a finite number of rectifiable Jordan curves we should extend to this case the known results of Caratheodory (1912), Lindelöf (1917), F. and M. Riesz (1916) and Lavrentiev (1936) for Jordan’s domains.

**Lemma 3.1.:** *Let D be a bounded domain in* ℂ *whose boundary components are Jordan curves,* 𝔻_{*} *be a bounded nondegenerate circular domain in* ℂ *and let ω* : *D* → 𝔻_{*} *be a conformal mapping. Then*

*(i)*

*ω can be extended to a homeomorphism of* $\overline{D}$ *onto* $\overline{{\mathbb{D}}_{*}}$;

*(ii)*

*arg [**ω*(*ζ*) – *ω*(*z*)] – arg [*ζ* – *z*] → *const as z* → *ζ whenever* ∂*D has a tangent at ζ* ∈ ∂*D*;

*(iii)*

*for rectifiable* ∂*D*, length *ω*^{−1} (*E*) = 0 *whenever* |*E*| = 0, *E* ⊂ ∂𝔻_{*};

*(iv)*

*for rectifiable* ∂*D*, |*ω*(*ℰ*)| = 0 *whenever* length *ℰ* = 0, *ℰ* ⊂ ∂*D*.

*Proof.* (i) Indeed, we are able to transform 𝔻_{*} into a simply connected domain 𝔻_{*} through a finite sequence of cuts. Thus, we come to the desired conclusion applying the Caratheodory theorems to simply connected domains 𝔻_{*} and *D*^{*} ≔ *ω*^{−1} (𝔻^{*}), see e.g. Theorem 9.4 in [10] and Theorem II.C.1 in [11].

(ii) In the construction from the previous item, we may assume that the point *ζ* is not the end of the cuts in *D* generated by the cuts in 𝔻_{*} under the extended mapping *ω*^{−1}. Thus, we come to the desired conclusion twice applying the Caratheodory theorems, the reflection principle for conformal mappings and the Lindelöf theorem for the Jordan domains, see e.g. Theorem II.C.2 in [11].

Points (iii) and (iv) are proved similarly to the last item on the basis of the corresponding results of F. and M. Riesz and Lavrentiev for Jordan domains with rectifiable boundaries, see e.g. Theorem II.D.2 in [11], and [12], see also the point III.1.5 in [13]. □

**Theorem 3.2.:** *Let D be a bounded multiply connected domain in* ℂ *whose boundary components are rectifiable Jordan curves and λ* : ∂*D* → ℂ, |*λ*(*ζ*)| ≡ 1, *and φ* : ∂*D* → ℝ *be measurable functions with respect to the natural parameter on* ∂*D. Then there exist multivalent analytic functions f* : 𝔻 → ℂ *with the infinite number of branches such that along any nontangential path*
$\underset{z\to \zeta}{\mathrm{lim}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Re}\text{\hspace{0.5em}}\overline{\{\lambda (\zeta )}\cdot f(z)\}=\phi (\zeta )\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}for\text{\hspace{0.17em}}a.e.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\zeta \in \partial D$(5)
*with respect to the natural parameters of the boundary components of D.*

*Proof.* This case is reduced to the case of a bounded nondegenerate circular domain 𝔻_{*} in the following way. First, there is a conformal mapping *ω* of *D* onto a circular domain 𝔻_{*}, see e.g. Theorem V.6.2 in [2]. Note that 𝔻_{*} is not degenerate because isolated singularities of conformal mappings are removable due to the well-known Weierstrass theorem, see e.g. Theorem 1.2 in [10]. Without loss of generality, we may assume that 𝔻_{*} is bounded.

By point (i) in Lemma 3.1 *ω* can be extended to a homeomorphisms of $\overline{D}$ onto $\overline{{\mathbb{D}}_{*}}$. If ∂*D* is rectifiable, then by point (iii) in Lemma 3.1 length *ω*^{−1} (*E*) = 0 whenever *E* ⊂ ∂𝔻_{*} with |*E*| = 0, and by (iv) in Lemma 3.1, conversely, |*ω*(*ℰ*)| = 0 whenever *ℰ* ⊂ ∂𝔻 with length *ℰ* = 0.

In the last case *ω* and *ω*^{−1} transform measurable sets into measurable sets. Indeed, every measurable set is the union of a sigma-compact set and a set of measure zero, see e.g. Theorem III(6.6) in [14], and continuous mappings transform compact sets into compact sets. Thus, a function *φ* : ∂*D* → ℝ is measurable with respect to the natural parameter on ∂*D* if and only if the function Φ = *φ* ∘ *ω*^{−1} : ∂𝔻_{*} → ℝ is measurable with respect to the natural parameter on ∂𝔻_{*}.

By point (ii) in Lemma 3.1, if ∂*D* has a tangent at a point *ζ* ∈ ∂*D*, then arg [*ω*(*ζ*) – *ω*(*z*)] – arg [*ζ* – *z*] → *const* as *z* → *ζ*. In other words, the conformal images of sectors in *D* with a vertex at *ζ* is asymptotically the same as sectors in 𝔻_{*} with a vertex at *w* = *ω*(*ξ*). Thus, nontangential paths in *D* are transformed under *ω* into nontangential paths in 𝔻_{*} and inversely. Finally, a rectifiable Jordan curve has a tangent a.e. with respect to the natural parameter and, thus, Theorem 3.2 follows from Theorem 2.1. □

In particular, choosing *λ* ≡ 1 in (5), we obtain the following statement.

**Proposition 3.3.:** *Let D be a bounded multiply connected domain in* ℂ *whose boundary components are rectifiable Jordan curves and let φ* : ∂*D* → ℝ *be measurable. Then there exist multivalent analytic functions f* : *D* → ℂ *with the infinite number of branches such that*
$\underset{z\to \zeta}{\mathrm{lim}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Re}f(z)=\phi (\zeta )\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}for\text{\hspace{0.17em}}\text{\hspace{0.17em}}a.e.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\zeta \in \partial D$(6)
*along any nontangential path with respect to the natural parameters of the boundary components of* ∂*D*.

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