An inequality for a two variable function

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The discussion focuses on proving the existence of a positive constant c such that F(u,v) is greater than or equal to c times the sum of |u|^{r+1} and |v|^{r+1}. Participants suggest analyzing the curve described by the expression in parentheses to understand its implications. The role of c is to establish a lower bound for F(u,v), ensuring it remains positive. The minimum value of the first expression is also a point of inquiry, as it relates to the overall inequality. Establishing these relationships is crucial for demonstrating the inequality holds under the given conditions.
amirmath
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Suppose that ##F(u,v)=a|u+v|^{r+1}+2b|uv|^{\frac{r+1}{2}}##, where ##a>1, b>0## and ##r\geq3.## How we can show that there exists a positive constant c such that
##
F(u,v)\geq c\Big( |u|^{r+1}+|v|^{r+1}\Big).
##
 
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You'd start by considering what sort of curve is described by the part in parentheses in the second relation.
What role does c play? Is there a minimum value that the first expression can take?
 

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