A mixed Hölder and Minkowski inequality

A Mixed Hölder and Minkowski Inequality

Hölder’s inequality states that ‖x‖p ‖y‖q − 〈x, y〉 ≥ 0 for any (x, y) ∈ Lp(Ω) × Lq(Ω) with 1/p + 1/q = 1. In the same situation we prove the following stronger chains of inequalities, where z = y|y|q−2: ‖x‖p ‖y‖q − 〈x, y〉 ≥ (1/p) [( ‖x‖p + ‖z‖p )p − ‖x + z‖p ] ≥ 0 if p ∈ (1, 2], 0 ≤ ‖x‖p ‖y‖q − 〈x, y〉 ≤ (1/p) [( ‖x‖p + ‖z‖p )p − ‖x + z‖p ] if p ≥ 2. A similar result holds for complex valued fun...

Hölder continuity of a parametric variational inequality

‎In this paper‎, ‎we study the Hölder continuity of solution mapping to a parametric variational inequality‎. ‎At first‎, ‎recalling a real-valued gap function of the problem‎, ‎we discuss the Lipschitz continuity of the gap function‎. ‎Then under the strong monotonicity‎, ‎we establish the Hölder continuity of the single-valued solution mapping for the problem‎. ‎Finally‎, ‎we apply these resu...

The Brunn-Minkowski-Type Inequality

Hölder and Minkowski type inequalities for pseudo-integral

There are proven generalizations of the Hölder's and Minkowski's inequalities for the pseudo-integral. There are considered two cases of the real semiring with pseudo-operations: one, when pseudo-operations are defined by monotone and continuous function g, the second semiring ([a, b], sup,), where is generated and the third semiring where both pseudo-operations are idempotent, i.e., È = sup an...

Another View on the Hölder Inequality

Every diagonal matrix D yields an endomorphism on the n-dimensional complex vector space. If one provides the n with Hölder norms, we can compute the operator norm of D. We define homogeneous weighted spaces as a generalization of normed spaces. We generalize the Hölder norms for negative values, this leads to a proof of an extended version of the Hölder inequality. Finally, we formulate this v...