Lower bounds for average sample size in the tests of goodness-of-fit and homogeneity

The problems of testing hypothesis $P=P_0$ on the distribution $P$ of random variable $\xi$ against the class of alternatives
$$ \mathscr P_1=\{P\colon \sup_A|P(A)-P_0(A)|\ge\Delta\} $$
and of testing hypothesis $P_1=P_2$ on the distributions $P_1$ and $P_2$ of independent random variables $\xi$ and $\eta$ against the class of alternatives
$$ \mathscr P_2=\{(P_1,P_2)\colon \sup_A|P_1(A)-P_2(A)|\ge\Delta\} $$
are considered. Lower bounds for average sample size which is sufficient for the acceptance of decision with guaranted restrictions $(\alpha,\beta)$ on the probabilities of errors are established. The asymptotical (for $\Delta\to 0$) efficiency of Kolmogorov and Smirnov tests with respect to obtained bounds is investigated.