On the existence of entire functions of boundedl-index andl-regular growth

We prove that, under certain conditions on a positive functionl continuous on 0, +∞], there exists an entire transcendental functionf of boundedl-index such that lnlnM _f(r)lnL(r),r→∞, whereM _f (r)=max {|f(z)|: |z|=r} andL(r)=∫ ₀ ^r l(t)dt. Ifl(r)=r ^p-1 forr≥1, 0<ρ<∞, then there exists an entire functionf of boundedl-index such thatM _f (r)≈r ^p . Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1166–1182, September, 1996.