ImprovementofoneofS´andor‘sinequalities Nicu¸sorMinculete

OCTOGON MATHEMATICAL MAGAZINE Vol. 17, No.1, April 2009, pp 288-290

ISSN 1222-5657, ISBN 978-973-88255-5-0, www.hetfalu.ro/octogon

288

Improvement of one of S´ andor‘s inequalities

Nicu¸sor Minculete³²

ABSTRACT.The objective of this paper is to present an improvement of S´andor‘s inequality

√σk(n)·σ1(n) σk−1

2 (n) ≤n^−(k4−l) ·ⁿ

k+l2 +1

2 , for anyn, k, l∈N^∗, whereσ_k(n)is the sum ofkth powers of divisors of n, so σ_k(n) =P

d|n

d^k.

INTRODUCTION

Letn be a positive integer,n≥1. We note with σ_k(n) the sum ofkth powers of divisors ofn, so, σk(n) =P

d|n

d^k, whence we obtain the following equalities: σ1(n) =σ(n) and σ0(n) =τ(n)−the number of divisors of n.

In [1], J. S´andor shows that pσk(n)·σl(n)

σk−l

2 (n) ≤n^−(k4−l) ·n^k+l2 + 1

2 , for anyn, k, l∈N^∗ (1) In [2] an inequality which is due to J.B. Diaz and F.T. Metcalf is proved, namely:

Lemma 1.1 Letn be a positive integer,n≥2. For every

a₁, a₂, ..., a_n∈R and for everyb₁, b₂, ..., b_n∈R^∗ with m≤ ^a_bⁱ_i ≤M and m, M ∈R, we have the following inequality:

Xn i=1

a²_i +mM Xn i=1

b²_i ≤(m+M) Xn

i=1

aibi. (2)

32Received: 25.03.2009

2000Mathematics Subject Classification. 11A25

Key words and phrases. The sum of the natural divisors of ⁿ, the sum of ^kth powers of divisors ofⁿ

Improvement of one of S´andor‘s inequalities 289 1. MAIN RESULT

Theorem 1.2. For everyn, k, l∈N withn≥2 and ^k−₂^l ∈N the following relation

pσ_k(n)·σ_l(n) σ^k−l

2 (n) ≤ n^l−4kσ_k(n) +n^k−4lσ_l(n) 2σ^k−l

2 (n) ≤n^−(k4−l) ·n^k+l2 + 1

2 , is true. (3) Proof. In the Lemma 1.1, making the substitutiona_i =q

d^k_i and b_i = √¹

d^l_i, whered_i is the divisor of n, for anyi= 1, τ(n). Since

1≤ ^a_biⁱ = q

d^k+l_i ≤n^k+l2 and a_ib_i =d

k−l

i2 , we take m= 1 and M =n^k+l2 . Therefore, inequality (2) becomes

τ(n)X

i=1

d^k_i +n^k+l2

τ(n)X

i=1

1 d^l_i ≤

1 +n^{k+l2 τ(n)}X

i=1

d

k−l

i2

which is equivalent to

σ_k(n) +n^k+l2 σ_l(n) n^l ≤

1 +n^k+l2 σk−l

2 (n) so that

σ_k(n) +n^k−2lσ_l(n)≤

1 +n^k+l2 σ^k−l

2 (n), (4)

for everyn, k, l∈N with n≥2.

The arithmetical mean is greater than the geometrical mean or they are equal, so for everyn, k, l∈N withn≥2, we have

q

n^k−2lσ_k(n)σ_l(n)≤ σ_k(n) +n^k−2lσ_l(n)

2 . (5)

Consequently, from the relations (4) and (5), we deduce the inequality pσ_k(n)σ_l(n)

σk−l

2 (n) ≤ n^l−4kσ_k(n) +n^k−4lσ_l(n) 2σk−l

2 (n) ≤n^−(k4−l) ·n^k+l2 +l

2 .

Remark Fork→k+ 2 andl→k we obtain the relation pσ_k+2(n)·σ_k(n)

σ(n) ≤

√1

nσ_k+2(n) +√

nσ_k(n)

2σ(n) ≤ 1

√n ·n^k+1+ 1

2 , (6)

290 Nicu¸sor Minculete for everyn, k∈N with n≥2.

Fork=l, we deduce another inequality which is due to S´andor, namely, σ_k(n)

τ(n) ≤ n^k+ 1

2 , (7)

for everyn, k∈N withn≥2.

REFERENCES

[1] S´andor, J., On Jordan’s Arithmetical Function, Gazeta Matematic˘a nr.

2-3/1993.

[2] Drimbe, M.O.,Inegalit˘at¸i. Idei ¸si metode, Editura GIL, Zal˘au, 2003.

“Dimitrie Cantemir” University of Bra¸sov E-mail: minculeten@yahoo.com