On invariance of the growth properties of entire function with respect to operator of \(\rho\)-integration (Q1333680)

Let \(f(z)= \sum^ \infty_ 0 f_ k z^ k\) be an entire function. By the Gelfond-Leont'ev \(\rho\)-integration operator, the author means \[ I_{\rho,\mu}: f\mapsto \sum^ \infty_ 0 {\Gamma({k\over s}+\mu)\over \Gamma({k+ 1\over \rho}+ \mu)} f_ k z^ k \] and by the infinite order \(\rho\)-integration with the characteristic function \(g(z) \left(= \sum^ \infty_{k= 1} g_ k z^ k\right)\), the operator \[ L^{\rho,\mu}_ g: f\mapsto \sum^ \infty_{k= 0} g_ k \Gamma\Biggl({k\over \rho}+ 1\Biggr) (I^ k_{\rho,\mu} f)(z). \] The author proves that, under natural restrictions on the entire functions \(f\), \(g\), the operator \(L^{\rho,\mu}_ g\) preserves the property of completely regular growth of the function \(f\) along a ray.