The maximum value of g(x) at [0, 1] is greater than the maximum value of f(x) at (0, +∞).

G(x) is certain, and the maximum value of G(x) on [0, 1] is G (0) = 2;

Therefore, the maximum value of f(x) at (0, +∞) should be less than 2.

That is f (x)

ax+lnx & lt； 2

ax & lt2-lnx

a & lt(2-lnx)/x

Let h (x) = (2-lnx)/x.

Then a < h(x)min.

h'(x)=(- 1-2+lnx)/x？ =(lnx-3)/x？

When 0

So h(x) in (0, e? In (e? , +∞) increasing;

So the minimum value of h(x) is h(e? )=(2-lne？ )/e？ =- 1/e？

Therefore, the value range of a is: a.

Have fun! I hope I can help you. If you don't understand, please ask. I wish you progress in your study! O(∩_∩)O