In this article we introduce the tamily ε(α), α ∈ C. of α-exponentially convex functions f as normalized univalent functions in the unit disk D such that e αf is convex univalent. Out study was suggested by a formal and concrete relationship between ε(α) and the class of convex functions of bounded type, defined by Professor A. W. Goodman. We also exhibit the connections to the class C(α) of normalized convex univalent functions which omit the balue -1/α. C-C(0) and to the class CH(α) of normalized convex univalent functions whose ranges lie mside a horizontal strip of width φ/α. Beside the description of some basic geometrical and analytic properties of the functions in ε(α). we are mainly interested in the determination of various Koebe domains or Koebe radii. We solve (up to certain constrained minimizations the Koebe radius and Koebe domain problem for C∩ε(α) and CH(α), and the Koebe domain problems for C(α) and ε(α) Because of technical difficulties this cannot be used to obtain the sharp (explicit) Koebe radius for ε(α). We have, however, complete numerical results. At least we can show that the Koebe radius for C∩ε(α) is larger than the one for ε(α), for small α