VTRMC 2017 Problem 5

Let $f ( x , y ) = ( x + y ) / 2 , g ( x , y ) = \sqrt { x y } , h ( x , y ) = 2 x y / ( x + y )$ , and let $$S = \{ ( a , b ) \in \mathrm { N } \times \mathrm { N } | a \neq b \text { and } f( a , b ) , g ( a , b ) , h ( a , b ) \in \mathrm { N } \}$$ where $\mathbb{N}$ denotes the positive integers. Find the minimum of $f$ over $S$ .