Is this formula, other than (2.38) in peskin's quantum field theory, also true?

Question

This is soft interlude question. I am rereading the Peskin & Schroeder's Quantum field theory, p.23, (2.38) and some question arises.

First, let's refer to Lorentz transformations for scalar fields in QFT --- Peskin and Schroder. There, the questioner asked that why the (2.37) and (2.38) in P&S are true and answered by Andrew McAddams.

Second, on the other hand, I tried to derive (2.38) in my own way as follows:

$1.$ $U(\Lambda) |p\rangle = U(\Lambda)(\sqrt{2E_p}a_p^{\dagger} |0 \rangle=\sqrt{2E_p} U(\Lambda)a_p^{\dagger} |0\rangle$

$2.$ $U(\Lambda)|p\rangle = |\Lambda p\rangle = \sqrt{2E_{\Lambda p}}a_{\Lambda p}^{\dagger} | 0 \rangle$

From 1 and 2 , we obtain $$U(\Lambda)a_p^{\dagger}= \sqrt{\frac{E_{\Lambda p}}{E_p}}a_{\Lambda p}^{\dagger}.$$

Does this argument also work? If not, where did I made mistake? Is there a point that I missing? If this is also true, why did the author express the relation (2.38) : $$ U(\Lambda) a^\dagger_p U^{-1}(\Lambda) = \sqrt{\frac{E_{\Lambda p}}{E_p}} a^\dagger_{\Lambda p}~? \tag{2.38}$$

Can anyone helps?