Your idea of involving an inverse sine function is good. I can use it to give an intuitive, geometric way to tackle the problem. But I think that this method is essentially rigorous, or at least can be made so.
i am going to write y=f(x) and y', y" for the derivatives. I want to show that at a point
on the graph of f, where the derivative is
, the inequality
must hold.
To start with, suppose that
and
are both positive. If the derivative stays at the constant value
then the function will increase steadily as shown by the line with the arrow in the diagram below, and it will soon exceed the maximum permitted value of 1. Therefore the derivative must decrease, so that the function increases more slowly. That means that the second derivative y" must be negative, and it must be sufficiently negative to stop the graph of the function crossing the line y=1 as it does in the diagram.
The inequality
shows that the most negative value y" can take is
So let's assume that y" takes this value, and we'll see whether this is sufficient to bend the graph of the function enough to stop it crossing the line y=1.
Thus we need to solve the differential equation
The solution is
, for some constant c. So
Integrating once more, we get
, where d is another constant. When x=c, this has the value 1+d. But y must not exceed the value 1, so we conclude that d must be negative, and thus
Put
to get the required condition
That deals with the case when
and
are both positive. A similar argument deals with the case when they are both negative. If one of them is positive and the other one is negative, you can turn the whole diagram upside down and replace the function f(x) by the function –f(–x). This function will have the feature that it and its derivative have the same sign at
So the previous argument will work again.