Yes, such exists, say works. Assume the contrary and consider the counterexample. Without loss of generality, , else multiple all 's to some so that this bacomes true, LHS is multiplied by , while RHS only by . So, we again get a counterexample. Denote , , . Since LHS does not exceed 2, we have (else RHS is too large). Hence for s.t. . Then we have , where , by some easy estimate. Finally, LHS equals and we just use estimate for small enough ( is small enough, since ).
Yes, such exists, say
Yes, such
exists, say
works. Assume the contrary and consider the counterexample. Without loss of generality,
, else multiple all
's to some
so that this bacomes true, LHS is multiplied by
, while RHS only by
. So, we again get a counterexample. Denote
,
,
. Since LHS does not exceed 2, we have
(else RHS is too large). Hence
for
s.t.
. Then we have
, where
,
by some easy estimate. Finally, LHS equals
and we just use estimate
for small enough
(
is small enough, since
).