Semochko N. Properties of logarithmic derivatives of analytic functions and solutions of complex differential equations of fractional order

In the thesis the Wiman-Valiron method for the fractional derivatives of entire functions is generalized. That is the behavior of fractional Riemann-Liouville derivative of order q>0 is investigated. The existence and the uniqueness of a solution of some fractional differential equation are proved. With the aid of the Wiman-Valiron method the order of the growth of solutions is found. We estimate a fractional integral of the logarithmic derivative of a meromorphic function. Also we investigate the fast growing entire solutions of linear differential equations. To do this we introduce a general scale to measure the growth of entire functions of infinite order including arbitrary fast growth. We described growth relations between entire coefficients and solutions of the linear differential equation in this scale in the complex plane. We investigate fast growing solutions of this equation in the unit disc.