The growth of entire functions in the terms of generalized orders - Hlova, T.Ya.; Filevych, P.V. (Carpathian Mathematical Publications, 2012)
Let Φ be a convex function on [x_0,+∞) such that Φ(x)x→+∞, x→+∞, f(z)=_n=0^∞ a_nz^n is a transcendental entire function, let M(r,f) be the maximum modulus of f and let ρ_Φ(f)=_r→ +∞M(r,f)Φ(r),c_Φ=_x→ +∞xΦ(x), d_Φ=_x→ +∞Φ'_+(x)Φ(x). It is proved that for every transcendental entire function f the generalized order ρ_Φ(f) is independent of the arguments of the coefficients a_n (or defined by the sequence (|a_n|)) if and only if the inequality d_Φc_Φ holds.