### A distortion theorem for bounded univalent functions.

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The paper is devoted to a class of functions analytic and univalent in the unit disk that are connected with an antigraphy ${e}^{i\phi}\overline{\omega}+i\rho {e}^{i\phi /2}$. Variational formulas and Grunsky inequalities are derived. As an application there are given some estimations in the considered class of functions.

Two-parameters extension of the family of typically-real functions is studied. The definition is obtained by the Stjeltjes integral formula. The kernel function in this definition serves as a generating function for some family of orthogonal polynomials generalizing Chebyshev polynomials of the second kind. The results of this paper concern the exact region of local univalence, bounds for the radius of univalence, the coefficient problems within the considered family as well as the basic properties...

In this article, we present a detailed study of the complex calculus of variations introduced in [M. Gondran: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I]. This calculus is analogous to the conventional calculus of variations, but is applied here to ${\mathbf{C}}^{n}$ functions in $\mathbf{C}$. It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions...