# Existence of a function satisfying some integral conditions

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):

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

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

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.