We investigate fast growing solutions of linear differential equations in the unit disc. For that we introduce a general scale to measure the growth of functions of infinite order including arbitrary fast growth. We describe the growth relations between entire coefficients and solutions of the linear differential equation f (n) + a n−1 (z)f (n−1) + . . . + a 0 (z)f = 0 in this scale and we investigate the growth of solutions where the coefficient of f dominates the other coefficients near a point on the boundary of the unit disc.

We generalize the Wiman-Valiron method for fractional derivatives proving that |z| q D q f (z) ∼ (ν(r, f)) q f (z) holds in a neighborhood of a maximum modulus point outside an exceptional set of values of |z| as |z| → ∞, where D q is the Riemann-Liouville fractional derivative of order q > 0, ν(r, f) is the central index of the Taylor representation of f. We use this result to find the precise value for the order of growth of solutions of a fractional differential equation.

Доведено iснування та єдинiсть розв'язку деякого дробового диференцiального рiвняння. За допомогою метода Вiмана-Валiрона знайдено порядок зростання розв'язку.