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We determine for several ranges of real parameters the order of starlikeness of the shifted Gauss hypergeometric function and we give some consequences of our results, in particular some mapping properties of the Carlson-Shaffer convolution operator.

Abstract. The spectrum of a class of self-inversive matrix polynomials is studied. It is shown that the characteristic values are semisimple and lie on the unit circle if the inner radius of an associated matrix polynomial is greater than 1 .Mathematics subject classification (2010): 15A33, 15A24, 15A57, 26C10, 30C15, 39A11.

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