On Cauchy–Schwarz type inequalities and applications to numerical radius inequalities

Abstract

In this article, we establish an improvement of the Cauchy-Schwarz inequality. Let x, y ∈ H, and let f : (0, 1) → R + be a well-defined function, where R + denote the set of all positive real numbers. ThenWe have applied this result to derive new and improved upper bounds for the numerical radius.

Refinement of the Cauchy-Schwartz inequality with refinements and generalizations of the numerical radius type inequalities for operators

“…The following inequalities are corollaries of (3.1). The following Lemma was given by Alomari [4]. Corollary 3.2.…”

“…Remark 4.6. Setting h = I, θ ∈ [0, 1] we obtain the inequality given by Alomari [4] (eq. 3.1), namely…”

“…for all p ≥ 1 and 0 ≤ α, β ≤ 1. For more results of this type, consult his paper [4]. Motivated by the previously described findings, the current work attempts to provide fresh refinements to the well-known Cauchy-Schwarz inequality in order to develop new numerical radius upper bounds of Hilbert space operators.…”

“…where P = T * T + T T * . Recently, Alomari [4] obtained refinements of (1.6), showing the following for T ∈ B(H ),…”

Further Accurate Numerical Radius Inequalities

“…In his recent work [4], Alomari refined the right-hand side of (3) and the recent results of Kittaneh and Moradi [5], as follows:…”