if $x-y =sqrt{x}-sqrt{y}$ with $xneq y$ then $(1+frac{1}{x})(1+frac{1}{y})geq 25$?
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let $xneq y$ be positive real numbers such that : $x-y= sqrt{x}-sqrt{y}$ , I have tried to prove this inequality $(1+frac{1}{x})(1+frac{1}{y})geq 25$ that i have created but i didn't got it. Attempt I have showed that: $(frac{1}{x}+frac{1}{y})geq frac{2}{sqrt{xy}}$ using this identity: $(sqrt{x}-sqrt{y})^2geq0$ , I also showed that : $frac{1}{xy}geq frac{1}{16}$ , Now I have used both result I have got the following inequality : $(1+frac{y}{x})(1+frac{x}{y})geq 25$ but not what i have claimed , any way ?
real-analysis inequality a.m.-g.m.-inequality
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edited Dec 31 '18 at 4:21