Някои нови неравенства между средните аритметично, геометрично, хармонично и квадратично
Some New Inequalities аmong the Arithmetic, Geometric, Harmonic and Quadratic Means

Summary/Abstract: Let n n n A , H , S be the arithmetic, harmonic and quadraticmeans respectivity for the positive real numbers 1 2 , ,..., n a a a . In this articlewe prove the following theorems:Theorem 1. For n = 3 we have(1 ). . 3n n n 3 A ≥ −λ H +λ S ⇔λ ≤ and(1 ). . 6n n n 3 A ≤ −λ H +λ S ⇔λ ≥ .Theotrem 2. For n = 4 we have(1 ). . 1n n n 2 A   H  S   and(1 ). . 3n n n 2 A ≤ −λ H +λ S ⇔λ ≥ .Theorem 3.For n = 5 we have (1 ). . 2 5n n n 5 A ≤ −λ H +λ S ⇔λ ≥ .Some open problems are proposed too.

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