I need help to prove the existence of a real function $h(x) \in C^1$ with condition that near zero $h(x) \sim \ln(x)$ and near infinity $\lim h(x)_{x \to \infty} = \infty$ such that following integrals are null (we have $0<a<\frac{1}{2}$):

$$\int_0^{\infty} h'(x)^{\frac{1}{2}} \; e^{(-a+ib) h(x)} dx =0 \;\; \, \;\; \,\;\; \, (1)$$

$$\int_0^{\infty} \big(h'(x)^{\frac{1}{2}} \; e^{(-a+ib) h(x)} -x^{-\frac{1}{2}-a+ib)}\big) x^{-\frac{1}{2}-a-ib)} dx =0 \;\; \, \;\; \,\;\; \, (2)$$

It is not difficult to show that an infinity of $h(x)$ satisfy (1): assuming $h(x)$ is strictly increasing we make the change of variable $y=h(x)$ and find that (1) becomes:

$$\int_{-\infty}^{\infty} G(y) \; e^{(-a+ib) y} dy =0 \;\; \, \;\; \,\;\; \, (3)$$

were $G$ is defined by $G(h(x))=h'(x)^{-\frac{1}{2}} $, so taking a $G$ function satisfying (3) (that we can easily construct by linear combination of different functions adjusted to have integral equal to zero) we deduce a function $h(x)$ exists as solution of the differential equation $G(h(x))=h'(x)^{-\frac{1}{2}} $.

But proving that some $h(x)$ function can satisfy (2) is far more difficult ! Even if commun sense tells us that such $h(x)$ functions exists. Any idea on how to treat this type of problem, any reference ? I am sure the same type of problem has already been treated but I was not able to find anything similar in literature.