We present a sufficient condition for the analyticity and univalence of the functions of the form \[ F(z)=\bigg(lphaıt^z_0\frac{f^lpha(u)}udu\bigg)^{1/lpha} \] using the method of subordination chains. They appear in the integral representation of $\alpha$-convex functions, a concept introduced by P. T. Mocanu.