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( x² - y² )² >= 0

x⁴ + y⁴ - 2 x² y² >= 0

x⁴ + y⁴ >= 2 x² y²

(x⁴ + y⁴) |y| >= 2 x² y² |y| >= 2 x² y³ ---- (1)

further, we also see that (x⁴ + y⁴) ( |y| + |x| ) >= 2 x² y³ + 2 x³ y²

Thus we see clearly that, from (1)

| (x² y³ ) / (x⁴ + y⁴) | <= |x| + |y|

x⁴ + y⁴ - 2 x² y² >= 0

x⁴ + y⁴ >= 2 x² y²

(x⁴ + y⁴) |y| >= 2 x² y² |y| >= 2 x² y³ ---- (1)

further, we also see that (x⁴ + y⁴) ( |y| + |x| ) >= 2 x² y³ + 2 x³ y²

Thus we see clearly that, from (1)

| (x² y³ ) / (x⁴ + y⁴) | <= |x| + |y|