Abstract: For $0< \alpha< \infty$, $N^\alpha$ is the class of meromorphic
functions $f$ defined in the unit disk $D$ satisfying $ \sup\limits_{z\in D}(1-|z|^2)^\alpha f^\#(z)<\infty$, where $f^\#(z)=|f'(z)|/(1+|f(z)|^2)$. Some new characterizations of the $\alpha $-normal functions are given for $0< \alpha< 1$. Characterizations are given in terms of the Bergman-Carleson measure conditions and the compact Bergman-Carleson conditions.

Key words: α-Normal functions, Normal functions, Meromorphic functions, Spherical derivative, Pseudohyperbolic disk