The conceptions of a harmonic measure introduced by R. Nevanlinna in [15] and a principal asymptotic value based on one nice result of F. Bagemihl [16] make possible with a great simplicity and generality to formulate the existence theorems for the Dirichlet and Riemann-Hilbert problems.

First of all, given a measurable set *E* ⊆ ∂𝔻 and a point *z* ∈ 𝔻, a *harmonic measure* of *E* at *z* relative to 𝔻 is the value at *z* of the bounded harmonic function *u* in 𝔻 with the boundary values 1 a.e. on E and a.e on ∂𝔻 \ *E*. In particular, by the mean value theorem for harmonic functions, the harmonic measure of *E* at 0 relative to 𝔻 is equal to |*E*|2*π*. In general, the geometric sense of the harmonic measure of *E* at *z*_{0} relative to 𝔻 is the angular measure of view of *E* from the point *z*_{0} in radians divided by 2*π*. Hence the harmonic measure on ∂𝔻 has also the corresponding probabilistic interpretation. The harmonic measure in domains *D* bounded by finite collections of Jordan curves is defined in a similar way.

Next, a Jordan curve generally speaking has no tangents. Hence we need a replacement for the notion of a nontangential limit. In this connection, recall Theorem 2 in [16], see also Theorem III.1.8 in [17], stating that, for any function $\Omega :\mathbb{D}\to \overline{\u2102}$, for all pairs of arcs *γ*_{1} and *γ*_{2} in 𝔻 terminating at *ζ* ∈ ∂𝔻, except a countable set of *ζ* ∈ ∂𝔻,
$C(\Omega ,\text{\hspace{0.17em}}\gamma 1)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cap \text{\hspace{0.17em}}\text{\hspace{0.17em}}C(\Omega ,\text{\hspace{0.17em}}\gamma 2)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\ne \varnothing $(7)
where *C*{Ω, *γ*) denotes the *cluster set of* Ω *at ζ along γ*, i.e.,
$C(\Omega ,\text{\hspace{0.17em}}\gamma )=\{w\in \overline{\u2102}\text{\hspace{0.17em}}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Omega ({z}_{n})\to w,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{n}\to \zeta ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{n}\in \gamma \}.$
Applying the Poincare mapping, branches of its inverse mapping and their boundary behavior, see e.g. Theorem VI.1 and Section VI.2 in [2], we extend this result to arbitrary domains *D* bounded by a finite number of Jordan curves, cf. the proof of Theorem 2.1.

Now, given a function $\Omega :D\to \overline{\u2102}$ and *ζ* ∈ ∂𝔻, denote by *P*(Ω, *ζ*) the intersection of all cluster sets *C*(Ω, *γ*) for arcs *γ* in *D* terminating at *ζ*. Later on, we call the points of the set *P*(Ω, *ζ*) *principal asymptotic values* of Ω at *ζ*. Note that, if Ω has a limit along at least one arc in *D* terminating at a point *ζ* ∈ ∂𝔻 with the property (7), then the principal asymptotic value is unique.

*Let D be a bounded multiply connected domain in* ℂ *whose boundary components are Jordan curves and let λ* : ∂*D* → ℂ, |*λ*(*ζ*)| ≡ 1, *and φ* : ∂𝔻 → ℝ *be measurable functions with respect to harmonic measures in D. Then there exist multivalent analytic functions f* : 𝔻 → ℂ *with the infinite number of branches such that*
$\underset{z\to \zeta}{\mathrm{lim}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Re}\text{\hspace{0.5em}}\overline{\{\lambda (\zeta )}\cdot f(z)\}=\phi (\zeta )\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}for\text{\hspace{0.17em}}a.e.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\zeta \in \partial D$(8)
*with respect to harmonic measures in D in the sense of the unique principal asymptotic value.*

*Proof.* By the reasons of the first item in the proof of Theorem 3.2, there is a conformal mapping *ω* of *D* onto a bounded nondegenerate circular domain 𝔻_{*} in ℂ. Set Λ = *λ* ∘ Ω and Φ = *φ* ∘ Ω where Ω ≔ *ω*^{−1} extended to ∂𝔻_{*} by point (i) in Lemma 3.1.

Note that harmonic measure zero is invariant under conformal mappings. Thus, arguing as in the third item of the proof to Theorem 3.2, we conclude that the functions Λ and Φ are measurable with respect to harmonic measures in 𝔻_{*}.

By Theorem 2.1 there exist multivalent analytic functions *F* : 𝔻_{*} → ℂ such that
$\underset{w\to \eta}{\mathrm{lim}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Re}\text{\hspace{0.5em}}\overline{\{\Lambda (\eta )}\cdot F(w)\}=\Phi (\eta )$
along any nontangential path to a.e. *η* ∈ ∂𝔻_{*}.

By the construction the functions *f* ≔ *F* ∘ *ω* are desired multivalent analytic solutions of (8) in view of the Bagemihl result. □

In particular, choosing *λ* ≡ 1 in (8), we obtain the following consequence.

*Let D be a bounded multiply connected domain in* ℂ *whose boundary components are Jordan curves and let φ* : ∂*D* → ℝ *be a measurable function with respect to harmonic measures in D. Then there exist multivalent analytic functions f* : *D* → ℂ *with the infinite number of branches such that*
$\underset{z\to \zeta}{\mathrm{lim}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Re}\text{\hspace{0.5em}}f(z)=\phi (\zeta )\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}for\text{\hspace{0.17em}}a.e.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\zeta \in \partial D$(9)
*with respect to harmonic measures in D in the sense of the unique principal asymptotic value.*

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