Generalized versions of reverse Young inequalities

Abstract

In this note, we obtain generalized versions of reverse Young inequalities as follows: For a 1 , a 2 , … , a n ∈ [ m , M ] a_{1},a_{2},\ldots ,a_{n}\in \left [ m,M\right ] with M ≥ m > 0 M\geq m>0 \[ v 1 a 1 + v 2 a 2 + ⋯ + v n a n ≤ S ( M m ) a 1 v 1 a 2 v 2 … a n v n v_{1}a_{1}+v_{2}a_{2}+\cdots +v_{n}a_{n}\leq S\left ( \frac {M}{m}\right ) a_{1}^{v_{1}}a_{2}^{v_{2}}\ldots a_{n}^{v_{n}} \] and \[ v 1 a 1 + v 2 a 2 + ⋯ + v n a n ≤ max a i ∈ [ m , M ] S ( a i a j ) L ( a i , a j ) + a 1 v 1 a 2 v 2 … a n v n v_{1}a_{1}+v_{2}a_{2}+\cdots +v_{n}a_{n}\leq \underset {a_{i}\in \left [ m,M\right ] }{\max }S\left ( \frac {a_{i}}{a_{j}}\right ) L\left ( a_{i,}a_{j}\right ) +a_{1}^{v_{1}}a_{2}^{v_{2}}\ldots a_{n}^{v_{n}} \] where S ( ⋅ ) S\left ( \cdot \right ) is Specht’s ratio, L ( a i , a j ) L\left ( a_{i,}a_{j}\right ) is logarithmic mean and v i ∈ [ 0 , 1 ] v_{i}\in \left [ 0,1\right ] such that v 1 + v 2 + ⋯ + v n = 1. v_{1}+v_{2}+\cdots +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.