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Austrian-Polish
1993 Austrian-Polish Competition
6
(\sqrt{a}+\sqrt{b})/2)^2 <=(a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b)/4 <=
(\sqrt{a}+\sqrt{b})/2)^2 <=(a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b)/4 <=
Source: Austrian Polish 1993 APMC
April 26, 2020
inequalities
algebra
Problem Statement
If
a
,
b
≥
0
a,b \ge 0
a
,
b
≥
0
are real numbers, prove the inequality
(
a
+
b
2
)
2
≤
a
+
a
2
b
3
+
a
b
2
3
+
b
4
≤
a
+
a
b
+
b
3
≤
(
a
2
/
3
+
b
2
/
3
2
)
3
\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^2\leq\frac{a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b}{4}\leq\frac{a+\sqrt{ab}+b}{3} \leq \sqrt{\left(\frac{a^{2/3}+b^{2/3}}{2}\right)^{3}}
(
2
a
+
b
)
2
≤
4
a
+
3
a
2
b
+
3
a
b
2
+
b
≤
3
a
+
ab
+
b
≤
(
2
a
2/3
+
b
2/3
)
3
For each of the inequalities, find the cases of equality.
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