I trust you give my application serious consideration. The verb "trust" is not used in combination with a bare infinitive. You have created a subjunctive form: I trust you give (I trust that you give) that is rarely if ever used in English. A subjunctive form would mean that there was some doubt in your trust. Better would be: I trust you to give my application serious consideration. or as Steve suggested using the future tense, which is more polite: I trust that you will give my application serious consideration. or I hope that you will give my application serious consideration......even more polite... You might want to stick with the tried and true universal formula: Thank you in advance for considering my application. No hoping or trusting needed! :)

" I trust that you will give my application serious consideration." sounds better

I trust you TO give my application serious consideration.

No... it think there is some mistake in this sentense which can only be pointed out when i know what u want to convey