For an algebroid function in the unit disk of finite lower order with a deficient value, we can estimate its growth order in terms of the convergence exponent of the points of the deficient value and other distinct values not lying on a radial system and the maximal difference of the arguments of adjacent rays.
"On the growth order of an algebroid function with radially distributed values in the unit disk." Tohoku Math. J. (2) 66 (3) 409 - 425, 2014. https://doi.org/10.2748/tmj/1412783205