On entire functions of exponential type

Le t J be a n en tire fun c ti o n a nd le t p. "" 1 and 1(1 , r)= {f:~ 1 f 11)(re iO) I "dO} 1/". for a U s uffic ie ntly la rge r, the n J is of ex pone nti a l type not exceeding. {2 log (l-t. ~) + 1 + log (2N) !} .. If thi s co ndition is re place d by re lated co nditi ons, th e n a lso is of expo ne nti a l t ype. An e ntire fun c tion f(z) is said to be of bounde d ind ex if and only if th ere exi sts a non-negative integer N (ind e pe nde nt of z) s uch thatO"'j ",N j!-k! (1.1) for all k and all z, and the smallest s uc h integer N is calle d the index off(z) ([1], [4] , [5]).1 It is known that a function of bounde d inde x N is of ex ponenti al type not exceedin g N+ 1 [6] but that a function of expon e ntial type need not be of bounde d inde x. In fac t any e ntire fun c tion havin g ze ros of arbitrarily large multipli city is not of bound e d index and th ere exist fun c tion s with simple zeros and of exponential type whic h are not of bounded index [8]. In a recent paper [2] Fred Gross considers interesting variations of condition (1.1) and proves the following THEOREM A: Let f be entire and C a positive constant. If there exists a positive integer N such that for k=O, 1,. . , N, f satisfies one ofthefollowing,for all z with l z I sufficiently large: