Extension of Cauchy-Schwarz's inequality

It has been known that the inequality

holds for all real values of a's and b's.
Then, can it be verified that the inequalities

also hold for all real a's, b's, c's and d's ? (They are of similar form as the original Cauchy-Schwarz inequality)
And eventually, can we prove that

is true for all positive integers m and real x's?
If it does not hold, please state the condition for which it does not hold and give counter-example.