A NOTE ON A PAPER OF R. C. BUCK

BY S. K. SINGH [D. S. College, Aiigarh (India)]

Received March 9, 1953 (Communicated by Dr. S. M. Shah, F.A.Sc.)

LET f ( z ) be an entire function of order p (0 < p < co) and let

l i m s u p l o g M ( r ) _ T limsup n ( r ) _ L inf. rP t ; inf. ~ P - - l '

r-~OO r-~OO

where M (r) is Max If(z) l a n d . (r) is the number of zeros off(z) in ] z l < r. I • l=r

(R. P. Boas, 1946) has proved that

t p T 0 )

and L ~< epT (2)

Recently (R. C. Buck, 1952) has proved that

l + eL <~ e2pT (3)

and with the help of (3) he proved that equality cannot hold simultaneously in (1) and (2). He further proves that if

1 = pT (4)

then L <~ (2.32 . . . . ) pT (5)

The purpose of this note is to see how far Buck's results can be improved. (S. M. Shah, 1948, lemma 2.1) has already proved an inequality better than (3), namely

L + l <~ epT (6) From (6) it is clear that if L : epT, then l = 0. We prove:

THEOREM.--(i) If (4) holds then

L = pT

(ii) and that equality cannot hold simultaneously in (1) and (6) and hence afortiori it can not hold simultaneously in (1) and (2) too.

Proof.--(i) If l = pT, then clearly L >~ pT, hence it is sufficient to prove that L ~ pT.

Now n (r) > (1 -- ,) rP for r t> r0 = r0 (e). 120

A Note on A Paper of R. C. Buck 121

Let k ~> 1 then r k l l P

f n (t) -7- dt ~< A + log M (rk I:p) 0

hence io+/+ ':" f ~ A 4- log M (rk 1/0)

0 to r

SO rkZ/P

(l -- ~) rP f l q- n (r) -t dt <~ A~ q- log M (rk x/p) P t

hence (I _T eo) rO + n (r) log k log M (rk~ o)

prPk ~< o (1) + rPk hence

1 L log k N <T

SO L < T p ( k - - 1)

log k

Put k = 1 + ~/, where ~ is a positive quantity tending to zero.

Then L<~Tp ~7

+ 0(~ =) hence

L ~<pT

Proof.--(ii) Let 1 = pT, then by the above L = pT

s o l + L = 2pT < epT.

Now let l + L = peT, then I will be less than pT for if it were equal to pT then l + L will have to be less than epT.

Finally, I thank Dr. S. M. Shah for suggesting the problem to me and helping me throughout the preparation of this paper.

REFERENCES

1. Boas, R . P . .. " Fundamental Sets of Entire Functions," Annals of Math., 1946, 47, 21-32.

2. Buck, R . C . . . " On the Distribution of the Zeros of an Entire Funct ion," J.LM.S., 1952, 16, 147-49.

3. Shah, S .M. . . " A Note on Uniqueness Sets for Entire Funct ions ," Proc. Ind. Acad. Sci., 1948, 28, 1-8,