ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5190 on D3312: Real power mean
Subresult of R2034: Weighted Hölder's inequality for real power means in the case of two real sequences
Weighted Cauchy-Schwarz inequality for two real sequences
Formulation 0
Let $x_1, y_1, \ldots, x_N, y_N \in [0, \infty)$ each be an D4767: Unsigned real number.
Let $\lambda_1, \ldots, \lambda_N \in [0, \infty)$ each be an D4767: Unsigned real number such that
(i) \begin{equation} \sum_{n = 1}^N \lambda_n = 1 \end{equation}
Then
(1) \begin{equation} \sum_{n = 1}^N \lambda_n x_n y_n \leq \left( \sum_{n = 1}^N \lambda_n x_n^2 \right)^{1 / 2} \left( \sum_{n = 1}^N \lambda_n y_n^2 \right)^{1 / 2} \end{equation}
(2) \begin{equation} \sum_{n = 1}^N \lambda_n x_n y_n = \left( \sum_{n = 1}^N \lambda_n x_n^2 \right)^{1 / 2} \left( \sum_{n = 1}^N \lambda_n y_n^2 \right)^{1 / 2} \quad \iff \quad \exists \, c \in [0, \infty) : \forall \, n = 1, \ldots, N : x^p_n = c y^q_n \text{ or } \forall \, n = 1, \ldots, N : y^q_n = c x^p_n \end{equation}
Subresults
▶ R5192: Cauchy-Schwarz inequality for two real sequences
Proofs
Proof 0
Let $x_1, y_1, \ldots, x_N, y_N \in [0, \infty)$ each be an D4767: Unsigned real number.
Let $\lambda_1, \ldots, \lambda_N \in [0, \infty)$ each be an D4767: Unsigned real number such that
(i) \begin{equation} \sum_{n = 1}^N \lambda_n = 1 \end{equation}