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The class p×p of matrix-valued functions c(z) holomorphic in the unit disk D = {z ∈ C : |z| < 1}, having order p, and satisfying Re c(z) ≥ 0 in D is considered, as well as its subclass p×pΠ of matrix-valued functions c(z) ∈ p×p that have a meromorphic pseudocontinuation c−(z) to the complement De = {z ∈ C : 1 < |z| ≤ ∞} of the unit disk with bounded Nevanlinna characteristic in De. For matrix-valued functions c(z) of class p×pΠ a representation as a block of a certain Jp,m-inner matrix-valued function θ(z) is obtained. The latter function has a special structure and is called the Jp,m-inner dilation of c(z). The description of all such representations is given. In addition, the following special Jp,m-inner dilations are considered and described: minimal, optimal, ∗-optimal, minimal and optimal, minimal and ∗-optimal. Also, Jp,m-inner dilations with additional properties are treated: real, symmetric, rational, or any combination of them under the corresponding restrictions on the matrix-valued function c(z). The results extend to the case where the open upper half-plane C+ is considered instead of the unit disk D. For entire matrix-valued functions c(z) with Re c(z) ≥ 0 in C+ and with Nevanlinna characteristic in C−, the Jp,m-inner dilations in C+ that are entire matrix-valued functions are also described. §

$J_{\lowercase {p},\lowercase {m}}$-inner dilations of matrix-valued functions that belong to the Carathéodory class and admit pseudocontinuation