Upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order and exponent of center

Statement
Suppose $$p$$ is a prime number. Suppose $$G$$ is a finite p-group of order $$p^n$$. Further, suppose that the center of $$G$$ has exponent $$p^k$$ (so the prime-base logarithm of exponent of the center is $$k$$). Then, the Schur multiplier is also a finite p-group and the derived subgroup and  Schur multiplier satisfy the following order inequality:

$$|G'||M(G)| \le p^{(n-k)(n + k-1)/2}$$