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Inegalități importante
1. a>1 ak−1<ak ∀k≥1
a∈(0,1) ak<ak−1 ∀k≥1
2. 0<a≤b ⇒(am−bm)(an−bn)≥0 ∀m,n∈N
3. a+1a≥2 (∀) a>0 a+1a≤−2 ∀ a<0.
7. a2+b2+c2≥ab+bc+ca\[∀\]a,b,c∈R
8. 3(a2+b2+c2)≥(a+b+c)2 ∀a,b,c∈R
9. a2+b2+c2a+b+c≥13(a+b+c) ∀a,b,c∈R
10. √a+b+c≥√33(√a+√b+√c)∀a,b,c≥0
11. (n−1)(a21+...+a2n)≥2(a1a2+...a1an+a2a3+...+an−1an)\)\(
13. an+bn2≥(a+b2)2,∀n∈N,a,b>0.
14. 0<ab<2⇒ab<a+rb+r,∀r>0.
1<ab⇒ab>a+rb+r,∀r>0
15. |x|≤a (a>0) ⇔−a≤x≤a.
16. |a±b|≤|a|+|b|\[,\]a,b∈R sauC.
17. |a1±a2±...±an|≤|a1|+...+|an| , in\[R\]sau\[C\].
18. ||a|−|b||≤|a−b|\[in\]R\[sau\]C.
19. 1n2=1n⋅n≤1(n−1)n=1n−1−1n
1n!<1(n−1)n=1n−1−1n
20. a,b∈Z ,m,n∈Z ,\[√mn∉Q\]⇒|ma2−nb2|≥1.
21. Numerele pozitive a,b,cpot fi lungimile laturilor unui triunghi dacă şi numai dacă ∃ x,y,z∈R∗+a.i a=y+z,\[b=x+z,\]c=x+y.
22. (ab)a−b≥1 a≠b ∀a,b>0,
23. a,b,c∈R∗+⇒a+bc+b+ca+c+ab≥6.
24. Dacăx1,...,xn≥0si x1+...+xn=kconstant atunci produsul x2⋅x2...xne maxim când x1=...=xn=kn.\)\(
