ON AN INEQUALITY THE SMARANDCHE Maohua
CONCERNING FUNCfION Le
Abstract Let a,n be positive integers. In we prove that S(a)S(c1) . .• S(d') ~n!(S(a);n¡ . Key words Smarandache function,inequaity.
this
paper
For any positive integer a, let S(a) be the Smarandache function. fu[l ],Bencze proposed the following problem. Problem. For any positive integers a and n, prove the inequality. n (1)
I1S(tI)~n!(S(a);n.
k=1 In this paper we completely solve this problem. We prove the following result Theorem. For any positive integers a and n, the inepuality (1) holds. Proof By [2,Theorem],we have S(ab)~S(a)+S(b),
for
any (2 )
for we
positive
a
b. It implies
and
that
S{cI)~kS(a),
any positive get n
(3)
integers integers
a
and
k. Therefore, by
n
I1S(cI) ~ I1 (kS(a)}=n!(S(a))IL. k=l k=1
Thus, the inequality (1 )is proved.
234
(2),
References [1]
M .Bencze, PP.l388,
Octogon Math.
Mag.,7(1999),2:149.
[2] M.-H.Le, An inequality conerning the Smarandache function,
Smarandache
125. Department of Mathemaics Zhanjiang
Normal College
Zhanjiang, Guangdong P.R
CHINA
235
Notions J.
9(1998),124-