Abstract:
In this note, we obtain generalized versions of reverse Young inequalities as follows: For a(1), a(2), ... , a(n) is an element of [m, M] with M >= m > 0
v(1)a(1) + v(2)a(2) + ... + v(n)a(n) <= S (M/m) a(1)(v1)a(2)(v2) ... a(n)(vn)
and
v(1)a(1) + v(2)a(2) + ... + v(n)a(n) <= max(ai is an element of[m, M]) S(a(i)/a(j)) L(a(i), a(j)) + a(1)(v1)a(2)(v2) ... a(n)(vn)
where S(.) is Specht's ratio, L (a(i), a(j)) is logarithmic mean and v(i) is an element of [0,1] such that v(1) + v(2) + ... + v(n) = 1.
Unlike the proof methods used in the articles on Young's inequality, the proofs of this study are obtained through first order conditions for constrained optimization problems.
