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 ).