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On a Geometric Theorem

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Leonard GIUGIUC and Bogdan SUCEAVA

In this paper we’ll give the admissible values of k for which
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(a + b + c + d) + kabcd ≥ 16 + k (1)

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1
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holds for all positive real numbers a, b, c and d satisfying a+b+c+d = + + + .
a b c d
From a + b + c + d = 1 + 1 + 1 + 1 we get
a b c d
abcd (a + b + c + d) abc + abd + acd + bcd
1 = and abcd = .
abc + abd + acd + bcd a + b + c + d
So (1) writes as

abc + abd + acd + bcd (16 + k) (a + b + c + d) abcd
2
(a + b + c + d) + k ≥ .
a + b + c + d abc + abd + acd + bcd
(2)
Firstly, we’ll find k ≥ 0 for which (2) holds for all a, b, c, d > 0.
We may assume WLOG in (2) that a + b + c + d = 4.

Via AM-GM we get 0 < abcd ≤ 1.

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Claim 1. The function f : [0, 1) → (0, 1] as f (t) = (1 − t) (1 + 3t) ∀t ∈ [0, 1) is
bijectively decreasing.

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Proof. As f (t) = −12t(1 − t) ≤ 0 ∀t ∈ [0, 1), we deduce that f is strictly
decreasing. On the other hand, f (0) = 1 and f (1 − 0) = 0; f is continuous, hence
it is well defined and bijective.
Claim 2. Let t ∈ [0, 1) be a fixed real number. We consider the positive numbers
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a, b, c, d such that a + b + c + d = 4 and abcd = (1 − t) (1 + 3t). Then
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min (abc + abd + acd + bcd) = 4(1 − t) (1 + 2t) .
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Professor, Colegiul Nat , ional ,,Traian”, Drobeta Turnu Severin, leonardgiugiuc@yahoo.com
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Professor, Department of Mathematics, California State University, bsuceava@fullerton.edu
This paper was presented at the 23-th Annual Conference of S.S.M.R., Pites , ti, 2019.
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