On the growth of a composition of entire functions - Sheremeta, M.M. (Carpathian Mathematical Publications, 2018)
Let γ be a positive continuous on [0, +∞) function increasing to +∞ and f and g be arbitrary entire functions of positive lower order and finite order. In order that for _r→+∞ M_f(g)(r)M_f({γ(r)})=+∞, M_f(r)={|f(z)|: |z|=r}, it is necessary and sufficient that (γ(r))/(r)→ 0 as r→+∞. This statement is an answer to the question posed by A.P. Singh and M.S. Baloria in 1991. Also in order that _r→+∞M_F(r) M_f({γ(r)})=0,F(z)=f(g(z)), it is necessary and sufficient that (γ(r))/(r)→ ∞ as r→+∞.