When m=2/3, the function g(x) has one and only one zero;

When 0 < m < 2/3, the function g(x) has two zeros;

When m≤0, the function g(x) has one and only one zero;

To sum up:

When m > 2/3, the function g(x) has no zero;

When m=2/3 or m≤0, the function g(x) has one and only one zero;

When 0 < m < 2/3, the function g(x) has two zeros;

(3 analysis: ∫ for any b > a > 0, [f (b)-f (a)]/(b-a)

Equivalent to f (b)-b < f (a)-a constant;

Let h (x) = f (x)-x = lnx+m/x-x (x > 0),

∴h(x) monotonically decreases at (0, +∞);

∫ h ′ (x) =1/x-m/x2-1≤ 0 always holds when (0, +∞),

∴m≥-x^2+x=-(x- 1/2)^2+ 1/4(x>0)，

∴m≥ 1/4；

For m= 1/4, h ′ (x) = 0 only holds when x= 1/2;

The range of ∴m is [1/4, +∞).